1. Integration by substitution

\[ \int 4 e^{-7 x} d x \] u = -7x
du/dx = -7
du = -7 dx
\[ = \int \frac{-4}{7} d u e^u \] \[ = \frac{-4}{7} \int e^u d u \] \[ = \frac{-4}{7} e^u + C \] \[ = \frac{-4}{7} e^{-7x} + C \]

2. Integration of 1/xn

\[ \int -\frac{3150}{t^4} - 220 \] \[ = \int -3150 t^{-4} - 220 \] \[ = -3150 \frac{t^{-3}}{-3} - 220 t + C = 6530 \] \[ = 1050 t^{-3} - 220 t + 5700 \]

3. Area of rectangles

1 + 3 + 5 + 7 = 16

4. Area between functions

curve(x^2 - 2*x - 2, -5, 5)
curve(x + 2, -5, 5, add=T, col="red")

Set the equations equal to each other to calculate where they intersect.

x^2 - 2x - 2 = x + 2
x^2 - 3x - 4 = 0
(x-4)*(x+1) = 0
x = (4, -1)

\[ \int_{-1}^4 (x + 2) - (x^2 - 2x -2) \] \[ = \int -x^2 + 3x + 4 \]

\[ = \frac{x^3}{3} + \frac{3 x^2}{2} + 4 x + C \] Plugging in 4 and -1 and subtracting:

4^3/3 + 3*4^2/2 + 4*4 - ((-1)^3/3 + 3*(-1)^2/2 + 4*(-1)) = 64.1666667

5. Business example

Unless there is information missing (no price or demand curve formula is given), selling all 110 irons in one order would minimize costs.

6. Integration by Parts

\[ \int ln(9x) x^6 dx \] \[ \int f(x) g'(x) dx = f(x) g(x) - \int f'(x) g(x) dx \] f(x) = ln(9x)
f’(x) = 1/x
g’(x) = x6
g(x) = x7/7
\[ ln(9x) \frac{x^7}{7} - \int \frac{1}{x} \frac{x^7}{7} dx \] \[ ln(9x) \frac{x^7}{7} - \frac{1}{7} \int x^6 dx \] \[ ln(9x) \frac{x^7}{7} - \frac{1}{7} \frac{x^7}{7} + C \] \[ \frac{x^7}{49}(7 ln(9x) - 1) + C \]

7. Definite Integral

\[ \int_1^{e^6} \frac{1}{6x} dx \] \[ \frac{1}{6} \int x^{-1} dx \] \[ \frac{ln x}{6} \] Now plugging in e6 and 1 and subtracting:

log(exp(6))/6 - log(exp(1))/6 = 0.8333333

It is not a probability distribution because the area is not 1.