Soal 1a.

menghitung \(E(x)\) dengan menggunakan metode Trapezoida dengan \(n=4\), metode Simpsons untuk \(n=4\) dan 4-point Gauss Quadrature tanpa menggunakan software apapun.
sehingga diperoleh keseluruhan hasil ditunjukkan pada Tabel 1.

Soal 1b.

Jika diketahui nilai eksak dari \[E(x)=\frac{b\Gamma(2+\frac{1}{2})\Gamma(b)}{\Gamma(1+\frac{1}{a}+b)}\] metode mana yang paling mendekati untuk menghitung \(E(x)\) dengan toleransi 0.0001? Gunakan R!
Nilai eksak dari \(E(x)\)

kemudian menghitung nilai \(E(x)\) menggunakan beberapa metode berikut

• Mendefinisikan fungsi

f<-function(x){
  15*x^5*(1-x^5)^2
}

Soal 1b (Trapezoidal \(n=4\))

library(pracma)
  trapezoid <- function(ftn, a, b, n = 100) {
    h <- (b-a)/n
    x.vec <- seq(a, b, by = h)
    f.vec <- sapply(x.vec, ftn)     # ftn(x.vec)
    Trap <- h*(f.vec[1]/2 + sum(f.vec[2:n]) + f.vec[n+1]/2)
    return(Trap)
  }
trapfunc1<-trapezoid(f,0,1)
trapfunc1

## [1] 0.710227

• dengan toleransi 0.0001

exact_value=0.710227
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  res_trap <- trapezoid(f,0,1,n = n)
  
  err <- abs(res_trap-exact_value)
  
  cat("n=",n,", result=",res_trap,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=0.6312869, error=0.07894012
## n=5, result=0.6738123, error=0.03641467
## n=6, result=0.6914928, error=0.01873424
## n=7, result=0.6997126, error=0.01051443
## n=8, result=0.7039057, error=0.006321303
## n=9, result=0.7062116, error=0.004015356
## n=10, result=0.7075597, error=0.002667293
## n=11, result=0.7083885, error=0.001838522
## n=12, result=0.7089199, error=0.001307139
## n=13, result=0.7092729, error=0.0009541148
## n=14, result=0.7095147, error=0.000712349
## n=15, result=0.7096846, error=0.0005423739
## n=16, result=0.7098069, error=0.0004201042
## n=17, result=0.7098966, error=0.0003303616
## n=18, result=0.7099637, error=0.0002633072
## n=19, result=0.7100146, error=0.0002124014
## n=20, result=0.7100538, error=0.0001731994
## n=21, result=0.7100844, error=0.0001426188
## n=22, result=0.7101085, error=0.0001184833
## n=23, result=0.7101278, error=9.923088e-05

diperoleh \[\int_{0}^{4}15x^{5}(1-x^{5})^{2}\approx0.7101278\] dengan error (\(\varepsilon\)=9.923088e-05) dan \(n=23\)

Soal 1b (Simpsons \(n=4\))

simpson_n <- function(ftn, a, b, n = 100) {
  n <- max(c(2*(n %/% 2), 4))
  h <- (b-a)/n
  x.vec1 <- seq(a+h, b-h, by = 2*h)       # ganjil
  x.vec2 <- seq(a+2*h, b-2*h, by = 2*h)   # genap
  f.vec1 <- sapply(x.vec1, ftn)   # ganjil
  f.vec2 <- sapply(x.vec2, ftn)   # genap
  S <- h/3*(ftn(a) + ftn(b) + 4*sum(f.vec1) + 2*sum(f.vec2))
  return(S)
}

simpfunc1<-simpson_n(f,0,1)
simpfunc1

## [1] 0.7102284

• dengan toleransi 0.0001

exact_value=0.710227
tol <-  0.0001
err <- 1
n = 4
while(err>tol){
  res_trap <- simpson_n(f,0,1,n = n)
  
  err <- abs(res_trap-exact_value)
  
  cat("n=",n,", result=",res_trap,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=0.7683974, error=0.05817036
## n=5, result=0.7683974, error=0.05817036
## n=6, result=0.7497082, error=0.03948124
## n=7, result=0.7497082, error=0.03948124
## n=8, result=0.728112, error=0.01788497
## n=9, result=0.728112, error=0.01788497
## n=10, result=0.7188088, error=0.008581832
## n=11, result=0.7188088, error=0.008581832
## n=12, result=0.7147289, error=0.004501893
## n=13, result=0.7147289, error=0.004501893
## n=14, result=0.712782, error=0.002555012
## n=15, result=0.712782, error=0.002555012
## n=16, result=0.711774, error=0.001546962
## n=17, result=0.711774, error=0.001546962
## n=18, result=0.7112144, error=0.0009873756
## n=19, result=0.7112144, error=0.0009873756
## n=20, result=0.7108852, error=0.0006581651
## n=21, result=0.7108852, error=0.0006581651
## n=22, result=0.7106819, error=0.0004548628
## n=23, result=0.7106819, error=0.0004548628
## n=24, result=0.7105511, error=0.0003240819
## n=25, result=0.7105511, error=0.0003240819
## n=26, result=0.710464, error=0.0002369803
## n=27, result=0.710464, error=0.0002369803
## n=28, result=0.7104042, error=0.0001772138
## n=29, result=0.7104042, error=0.0001772138
## n=30, result=0.7103621, error=0.0001351299
## n=31, result=0.7103621, error=0.0001351299
## n=32, result=0.7103318, error=0.0001048198
## n=33, result=0.7103318, error=0.0001048198
## n=34, result=0.7103096, error=8.255045e-05

diperoleh \[\int_{0}^{4}15x^{5}(1-x^{5})^{2}\approx0.7103096\] dengan error (\(\varepsilon\)=8.255045e-05) dan \(n=34\)

Soal 1b (4-point Gauss Quadrature)

menggunakan fungsi yang ditransformasi \[\int_{0}^{4}15(\frac{1}{2}t+\frac{1}{2})^{5}(1-\left(\frac{1}{2}\right)^{5})^{2}\]

ft <- function(t){
  (15/2)*((1/2*t)+1/2)^5*(1-((1/2*t)+1/2)^5)^2
  }
#orde 4
gl<-gaussLegendre(4,-1,1)
#mendefinisikan koefisien pembobot (w) dan gauss point (x)
Ci1<-gl$w
xi1<-gl$x
#menghitung integral
I<-sum(Ci1*ft(xi1))
I

## [1] 0.6792999

• dengan toleransi 0.0001

exact_value=0.710227
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  
  gL <- gaussLegendre(n = n,a = -1,1)
  
  Ci1 <- gL$w # koefisien
  xi1 <- gL$x # gauss point
  
  res_gl <- sum(Ci1 * ft(xi1))
  
  err <- abs(res_gl-exact_value)
  
  cat("n=",n,", result=",res_gl,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
}

## n=4, result=0.6792999, error=0.03092708
## n=5, result=0.707375, error=0.002851979
## n=6, result=0.7101422, error=8.477726e-05

diperoleh \[\int_{0}^{4}15(\frac{1}{2}t+\frac{1}{2})^{5}(1-\left(\frac{1}{2}\right)^{5})^{2}\approx0.7101422\] dengan error (\(\varepsilon\)=8.477726e-05) dan \(n=6\)

• 4-point Gauss Quadrature

menggunakan fungsi yang tidak ditransformasi
\[15x^{5}(1-x^{5})^{2}\]

f3<-function(x){
  15*x^5*(1-x^5)^2
}
gL<-gaussLegendre(4,0,1)
ci<-gL$w
xi<-gL$x
I1<-sum(ci*f3(xi))
I1

## [1] 0.6792999

• dengan toleransi 0.0001

exact_value=0.710227
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  
  gL <- gaussLegendre(n = n,a = 0,1)
  
  Ci <- gL$w # koefisien
  xi <- gL$x # gauss point
  
  res_gl <- sum(Ci * f3(xi))
  
  err <- abs(res_gl-exact_value)
  
  cat("n=",n,", result=",res_gl,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
}

## n=4, result=0.6792999, error=0.03092708
## n=5, result=0.707375, error=0.002851979
## n=6, result=0.7101422, error=8.477726e-05

diperoleh \[15x^{5}(1-x^{5})^{2}\approx0.7101422\] dengan error (\(\varepsilon\)=8.477726e-05) dan \(n=6\)

diketahui nilai eksak dari \(E(x)\) adalah 0.710227 sedangkan menggunakan metode Simpson \(n=4\) diperoleh nilai pendekatan 0.7103096 dengan errornya \(\varepsilon=8.26e-05\), dan error dari penerapan metode simpson tersebut merupakan error terkecil dibandingkan dengan error metode lainnya.

Metode trapezoidal

menghitung integral menggunakan user-defined function

f2<-function(x){
  2*exp(-2*x)
}

menghitung integral menggunakan fungsi trapezoid dengan \(n=4\)

trapezoid(f2,0,4,n=4)

## [1] 1.312595

menggunakan fungsi trapzfun

library("pracma")
trapzfun(f2,0,4,maxit=4)

## $value
## [1] 1.020405
## 
## $iter
## [1] 4
## 
## $rel.err
## [1] 0.06120913

menggunakan R hingga hasil integral mendekati nilai eksak dengan toleransi 0.00001

• menghitung nilai eksak

fungs<-function(x){
  2*(exp(-2*x))
}
integrate(fungs,0,4)

## 0.9996645 with absolute error < 1.1e-14

exact_value=0.9996645
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  res_trap <- trapezoid(f2,0,4,n = n)
  
  err <- abs(res_trap-exact_value)
  
  cat("n=",n,", result=",res_trap,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=1.312595, error=0.3129303
## n=5, result=1.204348, error=0.2046839
## n=6, result=1.143553, error=0.1438882
## n=7, result=1.106174, error=0.1065098
## n=8, result=1.081614, error=0.08194924
## n=9, result=1.064635, error=0.06497078
## n=10, result=1.05242, error=0.05275531
## n=11, result=1.043343, error=0.04367879
## n=12, result=1.036418, error=0.03675326
## n=13, result=1.031015, error=0.0313503
## n=14, result=1.026719, error=0.0270549
## n=15, result=1.023249, error=0.02358421
## n=16, result=1.020405, error=0.02074012
## n=17, result=1.018045, error=0.01838055
## n=18, result=1.016066, error=0.0164015
## n=19, result=1.01439, error=0.0147254
## n=20, result=1.012958, error=0.01329349
## n=21, result=1.011725, error=0.01206057
## n=22, result=1.010656, error=0.01099142
## n=23, result=1.009723, error=0.01005831
## n=24, result=1.008904, error=0.009239095
## n=25, result=1.00818, error=0.008515985
## n=26, result=1.007539, error=0.00787452
## n=27, result=1.006967, error=0.007302862
## n=28, result=1.006456, error=0.006791242
## n=29, result=1.005996, error=0.006331541
## n=30, result=1.005581, error=0.005916966
## n=31, result=1.005206, error=0.005541802
## n=32, result=1.004866, error=0.005201208
## n=33, result=1.004556, error=0.004891065
## n=34, result=1.004272, error=0.00460785
## n=35, result=1.004013, error=0.004348533
## n=36, result=1.003775, error=0.004110501
## n=37, result=1.003556, error=0.003891487
## n=38, result=1.003354, error=0.003689518
## n=39, result=1.003167, error=0.00350287
## n=40, result=1.002995, error=0.003330033
## n=41, result=1.002834, error=0.003169677
## n=42, result=1.002685, error=0.003020629
## n=43, result=1.002546, error=0.00288185
## n=44, result=1.002417, error=0.002752418
## n=45, result=1.002296, error=0.002631513
## n=46, result=1.002183, error=0.002518402
## n=47, result=1.002077, error=0.002412428
## n=48, result=1.001978, error=0.002313005
## n=49, result=1.001884, error=0.002219603
## n=50, result=1.001796, error=0.002131746
## n=51, result=1.001714, error=0.002049003
## n=52, result=1.001635, error=0.001970985
## n=53, result=1.001562, error=0.001897339
## n=54, result=1.001492, error=0.001827745
## n=55, result=1.001426, error=0.00176191
## n=56, result=1.001364, error=0.001699569
## n=57, result=1.001305, error=0.001640479
## n=58, result=1.001249, error=0.001584417
## n=59, result=1.001196, error=0.001531181
## n=60, result=1.001145, error=0.001480583
## n=61, result=1.001097, error=0.001432452
## n=62, result=1.001051, error=0.001386631
## n=63, result=1.001007, error=0.001342973
## n=64, result=1.000966, error=0.001301345
## n=65, result=1.000926, error=0.001261623
## n=66, result=1.000888, error=0.001223692
## n=67, result=1.000852, error=0.001187446
## n=68, result=1.000817, error=0.001152787
## n=69, result=1.000784, error=0.001119624
## n=70, result=1.000752, error=0.001087871
## n=71, result=1.000722, error=0.00105745
## n=72, result=1.000693, error=0.001028287
## n=73, result=1.000665, error=0.001000315
## n=74, result=1.000638, error=0.000973468
## n=75, result=1.000612, error=0.0009476878
## n=76, result=1.000587, error=0.0009229182
## n=77, result=1.000564, error=0.0008991072
## n=78, result=1.000541, error=0.000876206
## n=79, result=1.000519, error=0.0008541686
## n=80, result=1.000497, error=0.0008329523
## n=81, result=1.000477, error=0.0008125168
## n=82, result=1.000457, error=0.0007928242
## n=83, result=1.000438, error=0.0007738389
## n=84, result=1.00042, error=0.0007555275
## n=85, result=1.000402, error=0.0007378584
## n=86, result=1.000385, error=0.000720802
## n=87, result=1.000369, error=0.0007043303
## n=88, result=1.000353, error=0.0006884168
## n=89, result=1.000338, error=0.0006730365
## n=90, result=1.000323, error=0.000658166
## n=91, result=1.000308, error=0.000643783
## n=92, result=1.000294, error=0.0006298663
## n=93, result=1.000281, error=0.0006163961
## n=94, result=1.000268, error=0.0006033534
## n=95, result=1.000255, error=0.0005907204
## n=96, result=1.000243, error=0.00057848
## n=97, result=1.000231, error=0.0005666162
## n=98, result=1.00022, error=0.0005551136
## n=99, result=1.000208, error=0.0005439578
## n=100, result=1.000198, error=0.0005331349
## n=101, result=1.000187, error=0.0005226319
## n=102, result=1.000177, error=0.0005124362
## n=103, result=1.000167, error=0.000502536
## n=104, result=1.000157, error=0.00049292
## n=105, result=1.000148, error=0.0004835774
## n=106, result=1.000139, error=0.0004744979
## n=107, result=1.00013, error=0.0004656717
## n=108, result=1.000122, error=0.0004570896
## n=109, result=1.000113, error=0.0004487425
## n=110, result=1.000105, error=0.000440622
## n=111, result=1.000097, error=0.00043272
## n=112, result=1.00009, error=0.0004250287
## n=113, result=1.000082, error=0.0004175406
## n=114, result=1.000075, error=0.0004102487
## n=115, result=1.000068, error=0.0004031462
## n=116, result=1.000061, error=0.0003962266
## n=117, result=1.000054, error=0.0003894836
## n=118, result=1.000047, error=0.0003829112
## n=119, result=1.000041, error=0.0003765039
## n=120, result=1.000035, error=0.0003702561
## n=121, result=1.000029, error=0.0003641625
## n=122, result=1.000023, error=0.0003582181
## n=123, result=1.000017, error=0.0003524181
## n=124, result=1.000011, error=0.0003467579
## n=125, result=1.000006, error=0.0003412329
## n=126, result=1, error=0.000335839
## n=127, result=0.9999951, error=0.0003305719
## n=128, result=0.9999899, error=0.0003254278
## n=129, result=0.9999849, error=0.0003204029
## n=130, result=0.99998, error=0.0003154935
## n=131, result=0.9999752, error=0.000310696
## n=132, result=0.9999705, error=0.0003060072
## n=133, result=0.9999659, error=0.0003014237
## n=134, result=0.9999614, error=0.0002969424
## n=135, result=0.9999571, error=0.0002925604
## n=136, result=0.9999528, error=0.0002882747
## n=137, result=0.9999486, error=0.0002840824
## n=138, result=0.9999445, error=0.000279981
## n=139, result=0.9999405, error=0.0002759677
## n=140, result=0.9999365, error=0.0002720401
## n=141, result=0.9999327, error=0.0002681958
## n=142, result=0.9999289, error=0.0002644324
## n=143, result=0.9999252, error=0.0002607477
## n=144, result=0.9999216, error=0.0002571395
## n=145, result=0.9999181, error=0.0002536057
## n=146, result=0.9999146, error=0.0002501442
## n=147, result=0.9999113, error=0.0002467531
## n=148, result=0.9999079, error=0.0002434306
## n=149, result=0.9999047, error=0.0002401747
## n=150, result=0.9999015, error=0.0002369837
## n=151, result=0.9998984, error=0.0002338558
## n=152, result=0.9998953, error=0.0002307895
## n=153, result=0.9998923, error=0.0002277832
## n=154, result=0.9998893, error=0.0002248352
## n=155, result=0.9998864, error=0.000221944
## n=156, result=0.9998836, error=0.0002191083
## n=157, result=0.9998808, error=0.0002163266
## n=158, result=0.9998781, error=0.0002135976
## n=159, result=0.9998754, error=0.0002109198
## n=160, result=0.9998728, error=0.0002082921
## n=161, result=0.9998702, error=0.0002057133
## n=162, result=0.9998677, error=0.000203182
## n=163, result=0.9998652, error=0.0002006972
## n=164, result=0.9998628, error=0.0001982577
## n=165, result=0.9998604, error=0.0001958624
## n=166, result=0.999858, error=0.0001935102
## n=167, result=0.9998557, error=0.0001912002
## n=168, result=0.9998534, error=0.0001889313
## n=169, result=0.9998512, error=0.0001867026
## n=170, result=0.999849, error=0.0001845131
## n=171, result=0.9998469, error=0.0001823618
## n=172, result=0.9998447, error=0.000180248
## n=173, result=0.9998427, error=0.0001781708
## n=174, result=0.9998406, error=0.0001761292
## n=175, result=0.9998386, error=0.0001741225
## n=176, result=0.9998366, error=0.00017215
## n=177, result=0.9998347, error=0.0001702108
## n=178, result=0.9998328, error=0.0001683042
## n=179, result=0.9998309, error=0.0001664294
## n=180, result=0.9998291, error=0.0001645858
## n=181, result=0.9998273, error=0.0001627727
## n=182, result=0.9998255, error=0.0001609893
## n=183, result=0.9998237, error=0.0001592352
## n=184, result=0.999822, error=0.0001575095
## n=185, result=0.9998203, error=0.0001558117
## n=186, result=0.9998186, error=0.0001541413
## n=187, result=0.999817, error=0.0001524976
## n=188, result=0.9998154, error=0.00015088
## n=189, result=0.9998138, error=0.0001492881
## n=190, result=0.9998122, error=0.0001477212
## n=191, result=0.9998107, error=0.0001461789
## n=192, result=0.9998092, error=0.0001446606
## n=193, result=0.9998077, error=0.0001431658
## n=194, result=0.9998062, error=0.0001416941
## n=195, result=0.9998047, error=0.000140245
## n=196, result=0.9998033, error=0.000138818
## n=197, result=0.9998019, error=0.0001374127
## n=198, result=0.9998005, error=0.0001360286
## n=199, result=0.9997992, error=0.0001346653
## n=200, result=0.9997978, error=0.0001333224
## n=201, result=0.9997965, error=0.0001319995
## n=202, result=0.9997952, error=0.0001306962
## n=203, result=0.9997939, error=0.0001294122
## n=204, result=0.9997926, error=0.0001281469
## n=205, result=0.9997914, error=0.0001269002
## n=206, result=0.9997902, error=0.0001256715
## n=207, result=0.999789, error=0.0001244606
## n=208, result=0.9997878, error=0.0001232671
## n=209, result=0.9997866, error=0.0001220908
## n=210, result=0.9997854, error=0.0001209311
## n=211, result=0.9997843, error=0.000119788
## n=212, result=0.9997832, error=0.0001186609
## n=213, result=0.999782, error=0.0001175497
## n=214, result=0.999781, error=0.0001164541
## n=215, result=0.9997799, error=0.0001153737
## n=216, result=0.9997788, error=0.0001143083
## n=217, result=0.9997778, error=0.0001132575
## n=218, result=0.9997767, error=0.0001122212
## n=219, result=0.9997757, error=0.0001111991
## n=220, result=0.9997747, error=0.0001101908
## n=221, result=0.9997737, error=0.0001091962
## n=222, result=0.9997727, error=0.0001082151
## n=223, result=0.9997717, error=0.000107247
## n=224, result=0.9997708, error=0.000106292
## n=225, result=0.9997698, error=0.0001053496
## n=226, result=0.9997689, error=0.0001044197
## n=227, result=0.999768, error=0.0001035021
## n=228, result=0.9997671, error=0.0001025965
## n=229, result=0.9997662, error=0.0001017028
## n=230, result=0.9997653, error=0.0001008207
## n=231, result=0.9997644, error=9.995e-05

Metode Simpson

menghitung integral menggunakan user-defined function dengan \(n=4\)

simpson_n(f2,0,4,n=4)

## [1] 1.058815

menggunakan R hingga hasil integral mendekati nilai eksak dengan toleransi 0.0001

exact_value=0.9996645
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  res_simp <- simpson_n(f2,0,4,n = n)
  
  err <- abs(res_simp-exact_value)
  
  cat("n=",n,", result=",res_simp,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=1.058815, error=0.05915075
## n=5, result=1.058815, error=0.05915075
## n=6, result=1.014089, error=0.01442435
## n=7, result=1.014089, error=0.01442435
## n=8, result=1.00462, error=0.004955555
## n=9, result=1.00462, error=0.004955555
## n=10, result=1.001777, error=0.002112444
## n=11, result=1.001777, error=0.002112444
## n=12, result=1.000706, error=0.00104161
## n=13, result=1.000706, error=0.00104161
## n=14, result=1.000234, error=0.0005699298
## n=15, result=1.000234, error=0.0005699298
## n=16, result=1.000002, error=0.000337077
## n=17, result=1.000002, error=0.000337077
## n=18, result=0.9998762, error=0.0002117416
## n=19, result=0.9998762, error=0.0002117416
## n=20, result=0.999804, error=0.0001395486
## n=21, result=0.999804, error=0.0001395486
## n=22, result=0.9997601, error=9.563677e-05

Metode Gauss Quadrature

Pada bagian ini digunakan 2 alternatif untuk menerapkan metode Gauss Quadrature, yaitu menggunakan fungsi yang ditransformasi dan fungsi yang tidak ditransformasi

• Fungsi yang ditransformasi

f2_x<-function(x){
  4*exp(-4*(x+1))
}

digunakan orde n = 4

gl2<-gaussLegendre(n=4,-1,1)

berikut merupakan koefisien pembobot dan nilai titik Gauss (Gauss point)

#koefisien
c<-gl2$w
c

## [1] 0.3478548 0.6521452 0.6521452 0.3478548

#titik Gauss 
x<-gl2$x
x

## [1] -0.8611363 -0.3399810  0.3399810  0.8611363

menghitung integral

I_2a<-sum(c * f2_x(x))
I_2a

## [1] 0.9976267

menggunakan R hingga hasil integrasi mendekati nilai eksak dengan toleransi 0.0001

exact_value=0.9996645
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  
  gL <- gaussLegendre(n = n,a = -1,1)

 Ci <- gL$w # koefisien
 xi <- gL$x # gauss point
  
  res_gl <- sum(Ci * f2_x(xi))

  err <- abs(res_gl-exact_value)
  
  cat("n=",n,", result=",res_gl,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=0.9976267, error=0.002037806
## n=5, result=0.9995785, error=8.59847e-05

• Fungsi yang tidak ditransformasi

f_2n<-function(x){
  2*exp(-2*x)
}

digunakan orde n = 4

g1l2<-gaussLegendre(n=4,0,4)

berikut merupakan koefisien pembobot dan nilai titik Gauss (Gauss point)

#koefisien
c_22<-g1l2$w
c_22

## [1] 0.6957097 1.3042903 1.3042903 0.6957097

#titik Gauss 
x_22<-g1l2$x
x_22

## [1] 0.2777274 1.3200379 2.6799621 3.7222726

menghitung integral

I_2n<-sum(c_22 * f_2n(x_22))
I_2n

## [1] 0.9976267

menggunakan R hingga hasil integrasi mendekati nilai eksak dengan toleransi 0.0001

exact_value=0.9996645
tol <-  0.0001
err <- 1
n = 4

while(err>tol){
  
  gL <- gaussLegendre(n = n,a = 0,4)

 Ci <- gL$w # koefisien
 xi <- gL$x # gauss point
  
  res_gl <- sum(Ci * f_2n(xi))

  err <- abs(res_gl-exact_value)
  
  cat("n=",n,", result=",res_gl,", error=",err,"\n",sep = "")
  
  n=n+1
  if(n==1000){
    break
  }
  
}

## n=4, result=0.9976267, error=0.002037806
## n=5, result=0.9995785, error=8.59847e-05

Metode Monte-Carlo

berikut merupakan user-defined function dari Monte-Carlo

mc_integral <- function(ftn, a, b,m=m){
  #Membangkitkan x berdistribusi U(a,b)
  x <- runif(m,a,b)
  # Menghitung rata-rata dari output fungsi
  Gx <- ftn(x)
  Gx_m <- mean(Gx)
  theta.hat <- (b-a)*Gx_m
  return(theta.hat)
}

dilakukan simulasi untuk beberapa nilai \(m\)

set.seed(1)
mc3<-mc_integral(f2,0,4,1000)

set.seed(2)
mc4<-mc_integral(f2,0,4,10000)

set.seed(3)
mc5<-mc_integral(f2,0,4,100000)

set.seed(4)
mc6<-mc_integral(f2,0,4,1000000)

set.seed(5)
mc7<-mc_integral(f2,0,4,10000000)

menggunakan R hingga hasil integrasi mendekati nilai eksak dengan toleransi 0.0001

set.seed(7)
exact_value=0.9996645
tol <-  0.0001
err <- 1
m = 100

while(err>tol){
  
  MC <- mc_integral(f2,m = m,a = 0,4)

  err <- abs(MC-exact_value)
  
  cat("m=",m,", result=",MC,", error=",err,"\n",sep = "")
  
  m=m+1
  if(m==1000){
    break
  }
  
}

## m=100, result=0.8456942, error=0.1539703
## m=101, result=0.6679096, error=0.3317549
## m=102, result=0.8408358, error=0.1588287
## m=103, result=0.9315674, error=0.0680971
## m=104, result=1.363266, error=0.3636019
## m=105, result=0.6865527, error=0.3131118
## m=106, result=1.178121, error=0.178457
## m=107, result=0.9906003, error=0.009064238
## m=108, result=0.9915124, error=0.008152116
## m=109, result=1.046911, error=0.04724661
## m=110, result=0.9032224, error=0.09644213
## m=111, result=0.8105549, error=0.1891096
## m=112, result=1.253891, error=0.2542261
## m=113, result=1.104059, error=0.1043944
## m=114, result=0.9938322, error=0.005832302
## m=115, result=1.20134, error=0.2016751
## m=116, result=1.151963, error=0.1522981
## m=117, result=0.7583816, error=0.2412829
## m=118, result=1.175328, error=0.1756637
## m=119, result=1.197883, error=0.1982185
## m=120, result=0.8997218, error=0.09994271
## m=121, result=1.296386, error=0.2967215
## m=122, result=1.093542, error=0.09387741
## m=123, result=1.152097, error=0.1524323
## m=124, result=1.063742, error=0.06407746
## m=125, result=0.8550735, error=0.144591
## m=126, result=1.045843, error=0.04617897
## m=127, result=0.9947459, error=0.004918613
## m=128, result=0.8449821, error=0.1546824
## m=129, result=1.003357, error=0.003692776
## m=130, result=1.042859, error=0.04319499
## m=131, result=1.224965, error=0.225301
## m=132, result=1.114253, error=0.1145883
## m=133, result=0.932696, error=0.06696851
## m=134, result=0.7364727, error=0.2631918
## m=135, result=0.9118742, error=0.08779027
## m=136, result=1.15551, error=0.155845
## m=137, result=1.161365, error=0.1617005
## m=138, result=0.8616319, error=0.1380326
## m=139, result=1.09471, error=0.09504515
## m=140, result=1.025335, error=0.02567013
## m=141, result=0.8496915, error=0.149973
## m=142, result=0.9652415, error=0.03442298
## m=143, result=1.067638, error=0.06797317
## m=144, result=0.9827285, error=0.016936
## m=145, result=1.174144, error=0.1744799
## m=146, result=0.8899171, error=0.1097474
## m=147, result=0.8738986, error=0.1257659
## m=148, result=1.012254, error=0.01258957
## m=149, result=0.7928401, error=0.2068244
## m=150, result=0.7954266, error=0.2042379
## m=151, result=1.11289, error=0.1132257
## m=152, result=0.8198939, error=0.1797706
## m=153, result=0.8972993, error=0.1023652
## m=154, result=1.078297, error=0.07863249
## m=155, result=1.132065, error=0.1324009
## m=156, result=0.7744245, error=0.22524
## m=157, result=1.050125, error=0.05046057
## m=158, result=0.994814, error=0.0048505
## m=159, result=0.9123079, error=0.0873566
## m=160, result=1.153425, error=0.1537609
## m=161, result=0.8128669, error=0.1867976
## m=162, result=1.127386, error=0.1277216
## m=163, result=0.9605523, error=0.03911225
## m=164, result=0.844596, error=0.1550685
## m=165, result=0.9294164, error=0.07024807
## m=166, result=0.9503716, error=0.04929294
## m=167, result=1.070792, error=0.07112715
## m=168, result=1.01741, error=0.0177455
## m=169, result=1.024504, error=0.02483974
## m=170, result=0.8246924, error=0.1749721
## m=171, result=1.408419, error=0.4087543
## m=172, result=0.8724835, error=0.127181
## m=173, result=1.205583, error=0.205919
## m=174, result=1.027058, error=0.02739315
## m=175, result=0.9568572, error=0.04280734
## m=176, result=1.178548, error=0.1788838
## m=177, result=0.7251246, error=0.2745399
## m=178, result=1.144303, error=0.1446388
## m=179, result=0.9507004, error=0.0489641
## m=180, result=1.041863, error=0.0421985
## m=181, result=0.7562556, error=0.2434089
## m=182, result=0.9985434, error=0.001121098
## m=183, result=1.034053, error=0.03438841
## m=184, result=0.9361661, error=0.06349843
## m=185, result=1.001511, error=0.00184642
## m=186, result=0.9227552, error=0.07690926
## m=187, result=1.240969, error=0.2413044
## m=188, result=1.156233, error=0.1565681
## m=189, result=1.105468, error=0.1058039
## m=190, result=1.061974, error=0.06230936
## m=191, result=0.8035363, error=0.1961282
## m=192, result=1.154017, error=0.1543526
## m=193, result=1.081122, error=0.08145764
## m=194, result=0.8787267, error=0.1209378
## m=195, result=1.141705, error=0.1420404
## m=196, result=0.9829099, error=0.01675458
## m=197, result=0.8125748, error=0.1870897
## m=198, result=1.026303, error=0.02663884
## m=199, result=1.016367, error=0.01670288
## m=200, result=0.7636801, error=0.2359844
## m=201, result=0.7322229, error=0.2674416
## m=202, result=1.063346, error=0.06368125
## m=203, result=1.061505, error=0.06184002
## m=204, result=1.01754, error=0.01787567
## m=205, result=0.8563972, error=0.1432673
## m=206, result=1.030114, error=0.03044941
## m=207, result=1.089897, error=0.09023283
## m=208, result=1.102345, error=0.1026808
## m=209, result=0.850201, error=0.1494635
## m=210, result=0.944527, error=0.05513748
## m=211, result=1.017836, error=0.01817191
## m=212, result=1.016418, error=0.01675365
## m=213, result=0.9294097, error=0.07025485
## m=214, result=1.059475, error=0.0598103
## m=215, result=1.000778, error=0.001113919
## m=216, result=0.7664724, error=0.2331921
## m=217, result=1.074855, error=0.07519062
## m=218, result=1.168957, error=0.1692925
## m=219, result=1.169802, error=0.1701373
## m=220, result=0.8582211, error=0.1414434
## m=221, result=0.9038652, error=0.09579929
## m=222, result=1.196572, error=0.196908
## m=223, result=0.9681723, error=0.03149222
## m=224, result=0.9729182, error=0.02674634
## m=225, result=0.8523474, error=0.1473171
## m=226, result=1.025115, error=0.02545053
## m=227, result=0.9507521, error=0.04891241
## m=228, result=0.9608886, error=0.03877595
## m=229, result=0.9312856, error=0.06837885
## m=230, result=1.005791, error=0.006126264
## m=231, result=0.9370338, error=0.06263073
## m=232, result=0.8439682, error=0.1556963
## m=233, result=0.8879836, error=0.1116809
## m=234, result=1.017002, error=0.01733709
## m=235, result=0.9914539, error=0.008210622
## m=236, result=0.8219056, error=0.1777589
## m=237, result=1.010877, error=0.01121272
## m=238, result=1.1279, error=0.1282354
## m=239, result=1.052469, error=0.05280484
## m=240, result=1.057228, error=0.05756317
## m=241, result=0.7910828, error=0.2085817
## m=242, result=1.174572, error=0.1749073
## m=243, result=1.093872, error=0.09420761
## m=244, result=0.9340829, error=0.06558156
## m=245, result=0.9631768, error=0.03648773
## m=246, result=1.248173, error=0.2485084
## m=247, result=1.033737, error=0.03407278
## m=248, result=0.8542513, error=0.1454132
## m=249, result=1.001127, error=0.001462748
## m=250, result=1.177694, error=0.1780297
## m=251, result=1.08435, error=0.08468545
## m=252, result=0.8981527, error=0.1015118
## m=253, result=1.024281, error=0.02461686
## m=254, result=1.061132, error=0.0614677
## m=255, result=1.223849, error=0.2241845
## m=256, result=0.8372186, error=0.1624459
## m=257, result=1.160513, error=0.1608489
## m=258, result=1.051115, error=0.05145069
## m=259, result=0.9692317, error=0.0304328
## m=260, result=1.054215, error=0.05455016
## m=261, result=0.986543, error=0.01312154
## m=262, result=0.9359115, error=0.06375304
## m=263, result=0.8543392, error=0.1453253
## m=264, result=1.048247, error=0.04858251
## m=265, result=1.202545, error=0.2028802
## m=266, result=1.050419, error=0.05075438
## m=267, result=1.225066, error=0.2254019
## m=268, result=1.138751, error=0.1390866
## m=269, result=0.9030037, error=0.09666076
## m=270, result=0.9680384, error=0.0316261
## m=271, result=0.9340842, error=0.06558034
## m=272, result=0.902171, error=0.09749351
## m=273, result=0.9306653, error=0.06899923
## m=274, result=0.9917045, error=0.007959999
## m=275, result=0.9094673, error=0.09019719
## m=276, result=1.052893, error=0.05322891
## m=277, result=0.9626977, error=0.03696678
## m=278, result=1.040849, error=0.04118492
## m=279, result=1.047091, error=0.04742603
## m=280, result=0.979647, error=0.02001747
## m=281, result=0.9203999, error=0.07926458
## m=282, result=1.088777, error=0.08911256
## m=283, result=1.073058, error=0.07339339
## m=284, result=0.997521, error=0.002143525
## m=285, result=0.9586569, error=0.04100764
## m=286, result=1.011424, error=0.01175987
## m=287, result=0.8203676, error=0.1792969
## m=288, result=1.120633, error=0.1209684
## m=289, result=1.068944, error=0.06927953
## m=290, result=1.072874, error=0.07320903
## m=291, result=0.8796355, error=0.120029
## m=292, result=1.161152, error=0.1614874
## m=293, result=0.9529725, error=0.04669196
## m=294, result=0.9761532, error=0.02351127
## m=295, result=1.071691, error=0.07202653
## m=296, result=0.969741, error=0.02992345
## m=297, result=0.9226238, error=0.07704073
## m=298, result=0.9367927, error=0.06287179
## m=299, result=1.047926, error=0.04826104
## m=300, result=1.171003, error=0.1713381
## m=301, result=0.9825219, error=0.0171426
## m=302, result=0.9439774, error=0.0556871
## m=303, result=1.027264, error=0.02759923
## m=304, result=1.018853, error=0.01918871
## m=305, result=0.9700802, error=0.0295843
## m=306, result=0.8804304, error=0.1192341
## m=307, result=0.9474727, error=0.05219178
## m=308, result=0.8681381, error=0.1315264
## m=309, result=0.9708055, error=0.02885896
## m=310, result=0.9918334, error=0.007831066
## m=311, result=1.188951, error=0.1892869
## m=312, result=1.113977, error=0.114312
## m=313, result=0.9507976, error=0.04886686
## m=314, result=1.031368, error=0.03170333
## m=315, result=1.01392, error=0.01425575
## m=316, result=0.8249576, error=0.1747069
## m=317, result=0.9958653, error=0.003799209
## m=318, result=0.9731597, error=0.02650476
## m=319, result=0.9372423, error=0.06242217
## m=320, result=1.053264, error=0.05359984
## m=321, result=1.104955, error=0.1052908
## m=322, result=0.9802981, error=0.01936638
## m=323, result=1.081157, error=0.08149214
## m=324, result=1.005645, error=0.005980356
## m=325, result=0.8110531, error=0.1886114
## m=326, result=1.024631, error=0.02496689
## m=327, result=1.051175, error=0.05151029
## m=328, result=0.924252, error=0.07541253
## m=329, result=0.9718028, error=0.02786166
## m=330, result=0.9692663, error=0.0303982
## m=331, result=0.9949811, error=0.004683357
## m=332, result=1.102415, error=0.1027503
## m=333, result=0.9831408, error=0.01652366
## m=334, result=1.047371, error=0.04770605
## m=335, result=0.978333, error=0.02133147
## m=336, result=1.012015, error=0.01235038
## m=337, result=0.9555694, error=0.04409506
## m=338, result=1.040247, error=0.04058257
## m=339, result=1.082024, error=0.08235999
## m=340, result=1.125823, error=0.1261589
## m=341, result=1.001911, error=0.002246515
## m=342, result=1.014365, error=0.01470071
## m=343, result=0.8696977, error=0.1299668
## m=344, result=0.8522575, error=0.147407
## m=345, result=1.024429, error=0.02476432
## m=346, result=1.052419, error=0.05275474
## m=347, result=1.086641, error=0.08697686
## m=348, result=1.068367, error=0.06870282
## m=349, result=1.029276, error=0.02961101
## m=350, result=0.8934789, error=0.1061856
## m=351, result=1.079106, error=0.07944168
## m=352, result=1.022386, error=0.02272139
## m=353, result=1.007482, error=0.007817882
## m=354, result=1.084429, error=0.08476452
## m=355, result=1.199846, error=0.200182
## m=356, result=0.9515629, error=0.04810155
## m=357, result=0.9553185, error=0.04434604
## m=358, result=1.155558, error=0.1558935
## m=359, result=1.058959, error=0.05929484
## m=360, result=1.06235, error=0.06268517
## m=361, result=0.9822347, error=0.01742982
## m=362, result=0.945569, error=0.0540955
## m=363, result=0.9239392, error=0.07572533
## m=364, result=1.021463, error=0.02179844
## m=365, result=1.143162, error=0.1434975
## m=366, result=1.053596, error=0.05393165
## m=367, result=1.048035, error=0.0483707
## m=368, result=0.9670179, error=0.03264662
## m=369, result=0.9612723, error=0.03839225
## m=370, result=0.9037996, error=0.0958649
## m=371, result=1.038748, error=0.03908327
## m=372, result=0.8198674, error=0.1797971
## m=373, result=1.050011, error=0.05034694
## m=374, result=0.8923614, error=0.1073031
## m=375, result=0.8330909, error=0.1665736
## m=376, result=0.8546143, error=0.1450502
## m=377, result=0.9646004, error=0.03506411
## m=378, result=0.9028241, error=0.09684037
## m=379, result=1.060436, error=0.06077124
## m=380, result=1.204229, error=0.204564
## m=381, result=0.8975857, error=0.1020788
## m=382, result=1.109748, error=0.1100833
## m=383, result=1.145375, error=0.1457106
## m=384, result=0.990568, error=0.009096498
## m=385, result=0.8525374, error=0.1471271
## m=386, result=0.9251076, error=0.07455693
## m=387, result=1.0142, error=0.01453526
## m=388, result=1.010857, error=0.01119276
## m=389, result=1.133558, error=0.1338935
## m=390, result=1.003098, error=0.00343324
## m=391, result=0.9197089, error=0.07995562
## m=392, result=1.159748, error=0.1600833
## m=393, result=0.9113157, error=0.08834884
## m=394, result=0.8786231, error=0.1210414
## m=395, result=1.097232, error=0.09756753
## m=396, result=1.037906, error=0.0382417
## m=397, result=0.9464535, error=0.05321096
## m=398, result=0.9878968, error=0.01176769
## m=399, result=0.9625394, error=0.03712513
## m=400, result=1.077348, error=0.07768341
## m=401, result=1.145951, error=0.1462866
## m=402, result=1.140971, error=0.141307
## m=403, result=0.9650071, error=0.03465745
## m=404, result=0.9957315, error=0.003933
## m=405, result=1.041148, error=0.04148314
## m=406, result=0.9131183, error=0.08654624
## m=407, result=1.061683, error=0.06201825
## m=408, result=0.9888466, error=0.01081794
## m=409, result=1.121039, error=0.1213741
## m=410, result=0.9469386, error=0.05272592
## m=411, result=1.041224, error=0.04155967
## m=412, result=0.8998369, error=0.09982756
## m=413, result=0.9900195, error=0.009644996
## m=414, result=0.9816808, error=0.01798372
## m=415, result=1.059014, error=0.05934949
## m=416, result=0.923432, error=0.07623251
## m=417, result=1.178994, error=0.1793294
## m=418, result=1.113619, error=0.1139545
## m=419, result=1.068765, error=0.06910045
## m=420, result=1.096534, error=0.09686924
## m=421, result=1.165928, error=0.1662639
## m=422, result=1.05928, error=0.05961556
## m=423, result=0.9882172, error=0.01144728
## m=424, result=1.029108, error=0.02944331
## m=425, result=1.026185, error=0.0265202
## m=426, result=1.036569, error=0.03690422
## m=427, result=0.9853114, error=0.01435312
## m=428, result=1.078477, error=0.07881254
## m=429, result=1.033109, error=0.03344456
## m=430, result=1.078519, error=0.07885421
## m=431, result=1.079147, error=0.07948281
## m=432, result=0.909691, error=0.08997351
## m=433, result=0.9987686, error=0.0008958514
## m=434, result=1.103652, error=0.1039877
## m=435, result=0.99726, error=0.002404458
## m=436, result=0.7983345, error=0.20133
## m=437, result=1.080335, error=0.08067021
## m=438, result=1.033962, error=0.03429704
## m=439, result=0.9830137, error=0.01665084
## m=440, result=1.110939, error=0.1112748
## m=441, result=0.884954, error=0.1147105
## m=442, result=0.9968538, error=0.002810684
## m=443, result=1.10807, error=0.1084059
## m=444, result=1.134045, error=0.1343806
## m=445, result=1.051884, error=0.05221941
## m=446, result=1.021827, error=0.02216259
## m=447, result=0.8881708, error=0.1114937
## m=448, result=0.9773482, error=0.02231631
## m=449, result=0.9697325, error=0.02993203
## m=450, result=1.046888, error=0.04722392
## m=451, result=1.033355, error=0.03369072
## m=452, result=0.9420322, error=0.05763232
## m=453, result=0.9892637, error=0.01040079
## m=454, result=0.971196, error=0.02846848
## m=455, result=1.06748, error=0.0678155
## m=456, result=0.9623352, error=0.03732927
## m=457, result=0.9518534, error=0.0478111
## m=458, result=0.9456724, error=0.05399209
## m=459, result=1.067265, error=0.06760082
## m=460, result=0.9875177, error=0.01214682
## m=461, result=1.074525, error=0.0748604
## m=462, result=0.9776507, error=0.02201377
## m=463, result=1.077356, error=0.07769173
## m=464, result=0.9669692, error=0.03269525
## m=465, result=1.000216, error=0.0005519826
## m=466, result=1.022163, error=0.02249882
## m=467, result=1.140634, error=0.1409699
## m=468, result=0.858194, error=0.1414705
## m=469, result=1.093331, error=0.09366689
## m=470, result=1.022585, error=0.02292061
## m=471, result=0.9998539, error=0.0001893592
## m=472, result=1.033324, error=0.03365904
## m=473, result=1.062826, error=0.06316139
## m=474, result=0.8463796, error=0.1532849
## m=475, result=0.8831275, error=0.116537
## m=476, result=1.00798, error=0.008315628
## m=477, result=1.082965, error=0.08330075
## m=478, result=0.9662559, error=0.03340862
## m=479, result=0.8789776, error=0.1206869
## m=480, result=0.9900503, error=0.009614169
## m=481, result=0.9705945, error=0.02907004
## m=482, result=0.8895741, error=0.1100904
## m=483, result=0.9988804, error=0.0007841455
## m=484, result=0.9819914, error=0.0176731
## m=485, result=1.035126, error=0.03546167
## m=486, result=0.9034262, error=0.09623833
## m=487, result=1.04661, error=0.04694576
## m=488, result=0.8530415, error=0.146623
## m=489, result=1.083453, error=0.08378806
## m=490, result=1.170853, error=0.1711885
## m=491, result=1.115605, error=0.1159407
## m=492, result=1.096642, error=0.0969778
## m=493, result=1.023326, error=0.02366155
## m=494, result=0.8504106, error=0.1492539
## m=495, result=1.020992, error=0.02132732
## m=496, result=1.006618, error=0.006953369
## m=497, result=0.997196, error=0.002468453
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## m=500, result=1.030668, error=0.03100326
## m=501, result=1.015779, error=0.01611448
## m=502, result=0.9602169, error=0.03944757
## m=503, result=0.9216953, error=0.07796916
## m=504, result=1.07331, error=0.07364502
## m=505, result=1.002886, error=0.003221719
## m=506, result=1.192768, error=0.193103
## m=507, result=0.9523401, error=0.04732443
## m=508, result=0.9520593, error=0.04760522
## m=509, result=0.9431454, error=0.05651908
## m=510, result=0.9513526, error=0.04831187
## m=511, result=0.9694155, error=0.030249
## m=512, result=1.047652, error=0.04798747
## m=513, result=0.9548197, error=0.04484484
## m=514, result=1.074466, error=0.07480176
## m=515, result=0.7958718, error=0.2037927
## m=516, result=0.9070013, error=0.09266322
## m=517, result=1.100865, error=0.1012004
## m=518, result=1.026021, error=0.02635637
## m=519, result=0.9915892, error=0.008075305
## m=520, result=0.9484997, error=0.05116481
## m=521, result=0.9220254, error=0.07763908
## m=522, result=0.992893, error=0.006771514
## m=523, result=1.02692, error=0.02725585
## m=524, result=1.065407, error=0.065743
## m=525, result=0.9231029, error=0.07656162
## m=526, result=0.9508337, error=0.04883075
## m=527, result=1.017274, error=0.01760957
## m=528, result=0.7892332, error=0.2104313
## m=529, result=0.8330501, error=0.1666144
## m=530, result=0.9665645, error=0.03310003
## m=531, result=0.853583, error=0.1460815
## m=532, result=0.9680774, error=0.03158713
## m=533, result=1.026092, error=0.02642789
## m=534, result=0.9691047, error=0.03055983
## m=535, result=0.8718728, error=0.1277917
## m=536, result=1.093025, error=0.0933603
## m=537, result=1.039005, error=0.03934054
## m=538, result=1.021512, error=0.02184783
## m=539, result=1.006826, error=0.00716127
## m=540, result=1.036527, error=0.03686282
## m=541, result=1.031591, error=0.03192668
## m=542, result=1.091078, error=0.09141377
## m=543, result=1.067592, error=0.06792791
## m=544, result=1.005999, error=0.006334978
## m=545, result=0.9116786, error=0.0879859
## m=546, result=1.19571, error=0.1960457
## m=547, result=0.9863623, error=0.0133022
## m=548, result=0.9807926, error=0.01887194
## m=549, result=0.9478986, error=0.05176592
## m=550, result=0.9184671, error=0.08119742
## m=551, result=0.9308521, error=0.06881244
## m=552, result=0.8501347, error=0.1495298
## m=553, result=0.9779832, error=0.02168127
## m=554, result=0.986387, error=0.01327751
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## m=556, result=0.8864188, error=0.1132457
## m=557, result=1.021433, error=0.02176812
## m=558, result=0.971039, error=0.02862551
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## m=560, result=0.9155456, error=0.0841189
## m=561, result=0.9649615, error=0.03470302
## m=562, result=0.9589554, error=0.04070908
## m=563, result=1.00735, error=0.007685705
## m=564, result=0.8740384, error=0.1256261
## m=565, result=1.151744, error=0.1520797
## m=566, result=1.129926, error=0.1302612
## m=567, result=1.013892, error=0.0142278
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## m=572, result=1.046667, error=0.04700258
## m=573, result=0.9573068, error=0.04235765
## m=574, result=0.99394, error=0.005724546
## m=575, result=1.012272, error=0.01260735
## m=576, result=1.135263, error=0.1355981
## m=577, result=1.015136, error=0.01547138
## m=578, result=1.102903, error=0.1032384
## m=579, result=1.069705, error=0.07004084
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## m=582, result=0.850638, error=0.1490265
## m=583, result=1.007991, error=0.008326353
## m=584, result=1.049186, error=0.04952136
## m=585, result=1.048763, error=0.04909836
## m=586, result=1.018181, error=0.01851666
## m=587, result=1.07483, error=0.07516546
## m=588, result=0.9223899, error=0.07727457
## m=589, result=0.9309248, error=0.06873972
## m=590, result=1.08059, error=0.08092596
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## m=592, result=0.9439909, error=0.05567363
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## m=594, result=0.9848757, error=0.01478882
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## m=596, result=1.036431, error=0.03676662
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## m=598, result=1.003418, error=0.003753343
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## m=603, result=0.9354833, error=0.06418122
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## m=608, result=0.9853278, error=0.01433667
## m=609, result=0.9076056, error=0.0920589
## m=610, result=1.020664, error=0.02099982
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## m=707, result=1.145123, error=0.1454585
## m=708, result=0.9647619, error=0.0349026
## m=709, result=0.8759654, error=0.1236991
## m=710, result=1.002137, error=0.002472173
## m=711, result=0.8855336, error=0.1141309
## m=712, result=1.035603, error=0.03593811
## m=713, result=1.045193, error=0.04552815
## m=714, result=0.8528272, error=0.1468373
## m=715, result=1.013857, error=0.01419229
## m=716, result=0.9818183, error=0.01784624
## m=717, result=0.8812377, error=0.1184268
## m=718, result=0.905542, error=0.0941225
## m=719, result=1.017448, error=0.01778339
## m=720, result=1.007476, error=0.007811443
## m=721, result=0.9694365, error=0.030228
## m=722, result=0.9160163, error=0.08364818
## m=723, result=0.9023632, error=0.09730133
## m=724, result=1.058352, error=0.05868728
## m=725, result=0.9638187, error=0.03584584
## m=726, result=1.058986, error=0.05932117
## m=727, result=1.054312, error=0.05464784
## m=728, result=0.9669383, error=0.03272625
## m=729, result=0.9681781, error=0.03148643
## m=730, result=1.038854, error=0.03918958
## m=731, result=0.8965056, error=0.1031589
## m=732, result=0.953276, error=0.04638848
## m=733, result=1.040519, error=0.04085462
## m=734, result=0.9423728, error=0.05729173
## m=735, result=0.9383351, error=0.0613294
## m=736, result=1.063805, error=0.06414029
## m=737, result=0.9883057, error=0.01135877
## m=738, result=1.052088, error=0.05242318
## m=739, result=0.9951021, error=0.004562372
## m=740, result=1.125386, error=0.1257212
## m=741, result=0.857163, error=0.1425015
## m=742, result=1.061697, error=0.06203219
## m=743, result=1.004302, error=0.004637692
## m=744, result=0.920804, error=0.07886047
## m=745, result=1.055982, error=0.05631723
## m=746, result=0.9652902, error=0.03437434
## m=747, result=0.9799696, error=0.01969494
## m=748, result=1.04637, error=0.04670525
## m=749, result=0.9938175, error=0.005846989
## m=750, result=1.006854, error=0.007189328
## m=751, result=1.055929, error=0.05626407
## m=752, result=1.13843, error=0.1387654
## m=753, result=1.153277, error=0.1536128
## m=754, result=0.9422503, error=0.05741423
## m=755, result=1.059667, error=0.06000212
## m=756, result=1.090146, error=0.09048111
## m=757, result=1.037102, error=0.03743727
## m=758, result=1.054655, error=0.05499002
## m=759, result=1.006599, error=0.006934693
## m=760, result=0.93062, error=0.06904452
## m=761, result=0.9925683, error=0.00709622
## m=762, result=1.059688, error=0.06002301
## m=763, result=0.9348768, error=0.06478771
## m=764, result=1.050317, error=0.05065204
## m=765, result=1.019286, error=0.01962143
## m=766, result=0.9608358, error=0.03882866
## m=767, result=1.074088, error=0.0744231
## m=768, result=1.016731, error=0.01706699
## m=769, result=0.9913367, error=0.008327766
## m=770, result=1.044615, error=0.04495087
## m=771, result=1.018189, error=0.01852409
## m=772, result=1.062129, error=0.06246486
## m=773, result=0.9632297, error=0.03643482
## m=774, result=0.9871784, error=0.01248605
## m=775, result=0.9129893, error=0.08667516
## m=776, result=1.047004, error=0.04733916
## m=777, result=1.081587, error=0.08192206
## m=778, result=0.9323466, error=0.0673179
## m=779, result=1.027188, error=0.02752387
## m=780, result=1.040699, error=0.04103407
## m=781, result=0.8167227, error=0.1829418
## m=782, result=0.9561125, error=0.04355198
## m=783, result=0.9817213, error=0.01794323
## m=784, result=0.9629234, error=0.0367411
## m=785, result=0.8752729, error=0.1243916
## m=786, result=1.114253, error=0.1145882
## m=787, result=0.9334943, error=0.06617024
## m=788, result=0.9013175, error=0.09834702
## m=789, result=1.021675, error=0.02201021
## m=790, result=0.9495548, error=0.05010975
## m=791, result=0.9503667, error=0.04929775
## m=792, result=0.8857838, error=0.1138807
## m=793, result=0.9964721, error=0.003192405
## m=794, result=1.028041, error=0.02837655
## m=795, result=1.028469, error=0.0288046
## m=796, result=1.024706, error=0.02504111
## m=797, result=1.008736, error=0.009071853
## m=798, result=1.032477, error=0.03281268
## m=799, result=1.026854, error=0.02718912
## m=800, result=0.938552, error=0.06111245
## m=801, result=1.01191, error=0.01224508
## m=802, result=0.9335678, error=0.06609671
## m=803, result=1.031066, error=0.03140154
## m=804, result=0.9679085, error=0.03175604
## m=805, result=0.9895546, error=0.01010985
## m=806, result=0.9800431, error=0.01962143
## m=807, result=1.028923, error=0.02925834
## m=808, result=1.102386, error=0.1027216
## m=809, result=0.9577243, error=0.04194018
## m=810, result=1.010403, error=0.01073899
## m=811, result=0.9390771, error=0.06058735
## m=812, result=0.8675574, error=0.1321071
## m=813, result=1.104907, error=0.1052425
## m=814, result=0.9711691, error=0.02849537
## m=815, result=0.8659339, error=0.1337306
## m=816, result=0.9574364, error=0.04222806
## m=817, result=1.058824, error=0.05915949
## m=818, result=1.175047, error=0.1753829
## m=819, result=0.9368952, error=0.06276931
## m=820, result=1.045935, error=0.04627023
## m=821, result=0.9725205, error=0.02714404
## m=822, result=0.98398, error=0.01568452
## m=823, result=1.033069, error=0.03340494
## m=824, result=0.9665963, error=0.03306821
## m=825, result=0.9254465, error=0.07421802
## m=826, result=1.081272, error=0.0816074
## m=827, result=1.074146, error=0.07448132
## m=828, result=1.110742, error=0.1110776
## m=829, result=0.9559857, error=0.04367875
## m=830, result=0.9440496, error=0.05561492
## m=831, result=0.9499412, error=0.04972333
## m=832, result=1.101467, error=0.1018028
## m=833, result=1.026591, error=0.02692656
## m=834, result=1.033816, error=0.03415153
## m=835, result=1.044222, error=0.04455797
## m=836, result=1.006574, error=0.006909872
## m=837, result=0.9734278, error=0.02623672
## m=838, result=1.120855, error=0.1211907
## m=839, result=0.9710677, error=0.02859684
## m=840, result=1.103802, error=0.1041372
## m=841, result=0.9387366, error=0.06092787
## m=842, result=1.048979, error=0.049315
## m=843, result=1.007786, error=0.008121825
## m=844, result=1.150615, error=0.1509505
## m=845, result=0.967557, error=0.0321075
## m=846, result=0.9607508, error=0.0389137
## m=847, result=1.007988, error=0.008323723
## m=848, result=1.082782, error=0.0831173
## m=849, result=0.9873609, error=0.01230359
## m=850, result=1.067519, error=0.06785402
## m=851, result=0.9816908, error=0.01797371
## m=852, result=1.063703, error=0.06403844
## m=853, result=1.053705, error=0.05404073
## m=854, result=1.003863, error=0.00419818
## m=855, result=0.9935944, error=0.006070102
## m=856, result=0.9304783, error=0.06918615
## m=857, result=0.9174929, error=0.08217165
## m=858, result=0.9762307, error=0.02343376
## m=859, result=1.086351, error=0.08668614
## m=860, result=0.9804728, error=0.01919175
## m=861, result=0.9688006, error=0.03086385
## m=862, result=1.104611, error=0.1049463
## m=863, result=1.017049, error=0.01738447
## m=864, result=0.9535244, error=0.04614014
## m=865, result=0.9416291, error=0.05803535
## m=866, result=0.9586842, error=0.04098025
## m=867, result=0.9815666, error=0.01809788
## m=868, result=0.9639468, error=0.0357177
## m=869, result=1.043612, error=0.0439473
## m=870, result=1.112044, error=0.1123798
## m=871, result=1.03825, error=0.03858542
## m=872, result=1.031567, error=0.03190209
## m=873, result=1.076168, error=0.07650318
## m=874, result=1.046568, error=0.04690395
## m=875, result=1.034631, error=0.03496692
## m=876, result=0.9967401, error=0.002924396
## m=877, result=0.9723571, error=0.02730738
## m=878, result=0.9195997, error=0.08006481
## m=879, result=0.9471201, error=0.05254436
## m=880, result=1.097869, error=0.09820478
## m=881, result=1.077661, error=0.07799635
## m=882, result=0.9657005, error=0.03396396
## m=883, result=1.04338, error=0.04371503
## m=884, result=0.9685079, error=0.03115658
## m=885, result=0.9910917, error=0.008572756
## m=886, result=1.015203, error=0.01553853
## m=887, result=0.9138141, error=0.08585038
## m=888, result=0.9756089, error=0.02405563
## m=889, result=1.095055, error=0.09539027
## m=890, result=0.9952096, error=0.004454858
## m=891, result=1.116563, error=0.1168988
## m=892, result=1.081381, error=0.08171663
## m=893, result=1.107541, error=0.1078768
## m=894, result=0.9616287, error=0.0380358
## m=895, result=0.9408156, error=0.05884887
## m=896, result=0.9764982, error=0.02316631
## m=897, result=0.9605823, error=0.03908219
## m=898, result=1.098865, error=0.09920087
## m=899, result=0.9136219, error=0.08604259
## m=900, result=1.003796, error=0.004131205
## m=901, result=0.9756922, error=0.02397234
## m=902, result=0.9381288, error=0.06153568
## m=903, result=0.9330658, error=0.06659869
## m=904, result=0.9706244, error=0.02904005
## m=905, result=0.8894075, error=0.110257
## m=906, result=0.9597314, error=0.03993313
## m=907, result=0.9466729, error=0.05299161
## m=908, result=1.009268, error=0.009603703
## m=909, result=0.9369874, error=0.06267713
## m=910, result=1.030356, error=0.03069133
## m=911, result=0.9924779, error=0.007186647
## m=912, result=0.8585766, error=0.1410879
## m=913, result=0.9557469, error=0.04391756
## m=914, result=1.014355, error=0.01469042
## m=915, result=1.01875, error=0.01908518
## m=916, result=1.006007, error=0.006342597
## m=917, result=1.073417, error=0.07375245
## m=918, result=0.9626706, error=0.03699386
## m=919, result=0.9859225, error=0.01374198
## m=920, result=0.990186, error=0.009478469
## m=921, result=0.9798855, error=0.01977901
## m=922, result=0.8841907, error=0.1154738
## m=923, result=0.9325137, error=0.0671508
## m=924, result=0.9730188, error=0.02664572
## m=925, result=0.9721005, error=0.02756397
## m=926, result=0.9714422, error=0.02822226
## m=927, result=0.9116453, error=0.08801918
## m=928, result=0.9609804, error=0.03868412
## m=929, result=1.054899, error=0.05523404
## m=930, result=1.009752, error=0.01008779
## m=931, result=0.9835239, error=0.01614061
## m=932, result=0.9955125, error=0.004152022
## m=933, result=0.9789989, error=0.02066559
## m=934, result=0.9750573, error=0.02460716
## m=935, result=0.9782089, error=0.02145559
## m=936, result=0.9502746, error=0.04938991
## m=937, result=0.9524665, error=0.04719804
## m=938, result=1.084822, error=0.08515776
## m=939, result=1.016745, error=0.01708034
## m=940, result=0.9442267, error=0.05543781
## m=941, result=1.060784, error=0.06111976
## m=942, result=1.011881, error=0.01221659
## m=943, result=0.9804367, error=0.01922778
## m=944, result=0.9799873, error=0.01967723
## m=945, result=0.9546992, error=0.04496531
## m=946, result=0.9768827, error=0.02278178
## m=947, result=1.073066, error=0.07340178
## m=948, result=1.12804, error=0.1283751
## m=949, result=0.9798291, error=0.01983541
## m=950, result=1.022814, error=0.02314976
## m=951, result=1.081519, error=0.08185426
## m=952, result=0.9969911, error=0.002673434
## m=953, result=0.9793353, error=0.02032921
## m=954, result=0.9860894, error=0.01357507
## m=955, result=1.013606, error=0.01394102
## m=956, result=1.0793, error=0.07963594
## m=957, result=0.9940949, error=0.005569551
## m=958, result=1.082273, error=0.08260885
## m=959, result=1.022834, error=0.0231697
## m=960, result=1.018059, error=0.01839455
## m=961, result=0.9704497, error=0.02921482
## m=962, result=1.005628, error=0.005963166
## m=963, result=1.04725, error=0.04758581
## m=964, result=1.063345, error=0.06368057
## m=965, result=1.0072, error=0.007535687
## m=966, result=1.0315, error=0.03183537
## m=967, result=0.9737497, error=0.02591476
## m=968, result=0.9793677, error=0.02029682
## m=969, result=0.8993501, error=0.1003144
## m=970, result=0.9568376, error=0.04282686
## m=971, result=0.9874629, error=0.01220159
## m=972, result=0.974372, error=0.02529247
## m=973, result=1.036009, error=0.03634408
## m=974, result=0.9645243, error=0.03514022
## m=975, result=0.9973148, error=0.002349658
## m=976, result=1.017938, error=0.0182731
## m=977, result=0.9593342, error=0.04033031
## m=978, result=1.030423, error=0.03075898
## m=979, result=0.9830467, error=0.01661775
## m=980, result=1.025128, error=0.0254634
## m=981, result=1.011256, error=0.01159159
## m=982, result=0.9756371, error=0.0240274
## m=983, result=0.9385458, error=0.06111872
## m=984, result=1.046402, error=0.04673777
## m=985, result=1.009362, error=0.009697253
## m=986, result=1.07876, error=0.07909588
## m=987, result=1.003396, error=0.003731448
## m=988, result=1.088829, error=0.08916461
## m=989, result=0.9598195, error=0.03984497
## m=990, result=0.9786421, error=0.02102236
## m=991, result=0.9859783, error=0.01368624
## m=992, result=0.9478837, error=0.05178085
## m=993, result=1.03811, error=0.03844552
## m=994, result=0.9780726, error=0.02159193
## m=995, result=1.026526, error=0.02686119
## m=996, result=1.000131, error=0.0004668655
## m=997, result=1.051349, error=0.05168493
## m=998, result=1.043586, error=0.04392133
## m=999, result=1.031797, error=0.03213219

Kesimpulan

Berdasarkan hasil trial untuk masing-masing metode diperoleh ringkasan hasil yang disajikan dalam tabel berikut

Berdasarkan keempat metode, diperoleh nilai pendekatan terkecil yaitu 0.9995785 menggunakan metode 4-point Gauss Quadrature. Nilai pendekatan tersebut juga merupakan nilai yang paling mendekati nilai eksak dengan error = 0.0000860. Oleh karena itu metode integrasi numerik yang paling baik yang dapat digunakan pada kasus ini adalah metode 4-point Gauss Quadrature.