7.1 Integration by Parts

1. 分部積分 (Integration by Parts)：不定積分(Indefinite Integrals)

\[\int f(x)g'(x)\;dx=f(x)g(x)-\int g(x)f'(x)\;dx\]

2. 代換法則 (Substitution Rule)

\[\int u\;dv=uv-\int v\;du\]

3. 指數和正弦函數的積分

\[\int e^x\sin x\;dx=-e^x\cos x+\int e^x\cos x\;dx\]

4. 指數和餘弦函數的積分

\[\int e^x\cos x\;dx=e^x\sin x-\int e^x\sin x\;dx\]

5. 分部積分 (Integration by Parts)：定積分(Definite Integrals)

\[\int_{a}^{b} f(x)g'(x)\;dx=\left. f(x)g(x)\right]_{a}^{b}-\int_{a}^{b} g(x)f'(x)\;dx\]

6. 遞迴公式 (Reduction Formulas)

\[\int \sin^n x\;dx=-\frac{1}{n}\cos x\;\sin^{n-1} x+\frac{n-1}{n}\int \sin^{n-2}x\;dx\]

7.2 Trigonometric Integrals

1. 正弦和餘弦冪次的積分 (Integrals of Powers of Sine and Cosine)

\[\int \sin^m x\;\cos^n x\;dx\]

     \((1)\;n\,為奇數\,(n=2k+1)\)

\[\int \sin^m x\;\cos^{2k+1} x\;dx=\int \sin^m x\;(\cos^2 x)^k\;\cos x\;dx=\int \sin^m x\;(1-\sin^2 x)^k\;\cos x\;dx\]

     \((2)\;m\,為奇數\,(m=2k+1)\)

\[\int \sin^{2k+1} x\;\cos^n x\;dx=\int (\sin^2 x)^k\;\cos^n x\;\sin x\;dx=\int (1-\cos^2 x)^k\;\cos^n x\;\sin x\;dx\]

2. 正割和正切冪次的積分 (Integrals of Powers of Secant and Tangent)

\[\int \tan^m x\;\sec^n x\;dx\]

     \((1)\;n\,為偶數\,(n=2k,\, k\ge 2)\)

\[\int \tan^m x\;\sec^{2k} x\;dx=\int \tan^m x\;(\sec^2 x)^{k-1}\;\sec^2 x\;dx=\int \tan^m x\;(1+\tan^2 x)^{k-1}\;\sec^2 x\;dx\]

     \((2)\;m\,為奇數\,(m=2k+1)\)

\[\int \tan^{2k+1} x\;\sec^n x\;dx=\int (\tan^2 x)^k\;\sec^{n-1} x\;\sec x\;\tan x\;dx=\int (\sec^2 x-1)^k\;\sec^{n-1} x\;\sec x\;\tan x\;dx\]

     The LibreTexts libraries Mathematics
     https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(Apex)/06%3A_Techniques_of_Integration/6.03%3A_Trigonometric_Integrals
     Paul’s Online Notes https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithTrig.aspx
     Simon Fraser University
     https://www.sfu.ca/math-coursenotes/Math%20158%20Course%20Notes/sec_Powers_of_trigonometric_functions.html


3.

\[\int \sec x\;dx=\ln |\sec x+\tan x|\]

4. 和差角公式

     \((1)\;\cos \;(\alpha-\beta)=\cos\alpha\;\cos\beta+\sin\alpha\;\sin\beta\)
     \((2)\;\cos \;(\alpha+\beta)=\cos\alpha\;\cos\beta-\sin\alpha\;\sin\beta\)
     \((3)\;\sin \;(\alpha+\beta)=\sin\alpha\;\cos\beta+\cos\alpha\;\sin\beta\)
     \((3)\;\sin \;(\alpha-\beta)=\sin\alpha\;\cos\beta-\cos\alpha\;\sin\beta\)


     The LibreTexts libraries Mathematics
     https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(OpenStax)/07%3A_Trigonometric_Identities_and_Equations/7.02%3A_Sum_and_Difference_Identities


5. 積化和差公式

     \((1)\;\sin A\;\cos B=\frac {1}{2}\left[\sin (A-B)+\sin (A+B) \right ]\)
     \((2)\;\sin A\;\sin B=\frac {1}{2}\left[\cos (A-B)-\cos (A+B) \right ]\)
     \((3)\;\cos A\;\cos B=\frac {1}{2}\left[\cos (A-B)+\cos (A+B) \right ]\)


7.3 Trigonometric Substitution

三角函數代換表
運算式 代換式 恆等式
\(\sqrt{a^2-x^2}\) \(x=a\sin\theta,\quad-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}\) \(1-\sin^2\theta = \cos^2\theta\)
\(\sqrt{a^2+x^2}\) \(x=a\tan\theta,\quad-\frac{\pi}{2}<\theta<\frac{\pi}{2}\) \(1+\tan^2\theta = \sec^2\theta\)
\(\sqrt{x^2-a^2}\) \(x=a\sec\theta,\quad0\le\theta<\frac{\pi}{2}\,or\,\pi\le\theta\le\frac{3\pi}{2}\) \(\sec^2\theta-1 = \tan^2\theta\)


7.4 Integration of Rational Functions by Partial Fractions

部分分式 (Partial Fractions)

\[f(x)=\frac{P(x)}{Q(x)}=S(x)+ \frac{R(x)}{Q(x)}\]

1. \(Q(x)\) 有相異一次因式 (distinct linear factors)

\[\frac{R(x)}{Q(x)}=\frac{A_1}{a_1x+b_1}+\frac{A_2}{a_2x+b_2}+ \cdots+\frac{A_k}{a_kx+b_k}\]

2. \(Q(x)\) 有重複的一次因式

\[\frac{A_1}{a_1x+b_1}+\frac{A_2}{(a_1x+b_1)^2}+\cdots+\frac{A_r}{(a_1x+b_1)^r}\]

3. \(Q(x)\) 有不可約二次因子 (irreducible quadratic factors)

\[\frac{Ax+B}{ax^2+bx+c}\]

4. \(Q(x)\) 有重複的不可約二次因子

\[\frac{A_1x+B_1}{ax^2+bx+c}+\frac{A_2x+B_2}{(ax^2+bx+c)^2}+\cdots+\frac{A_rx+B_r}{(ax^2+bx+c)^r}\]

The LibreTexts libraries Mathematics https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(OpenStax)/09%3A_Systems_of_Equations_and_Inequalities/9.04%3A_Partial_Fractions


7.7 Improper Integrals

1. 第一型瑕積分 (Improper Integral) 的定義

     \((1)\;若\;f(x)\;在\;[a,\infty)\;連續，則\;\int_{a}^{\infty} f(x)\;dx=\displaystyle\lim_{t \to \infty}\int_{a}^{t} f(x)\;dx\)
     \((2)\;若\;f(x)\;在\;(-\infty,b]\;連續，則\;\int_{-\infty}^{b} f(x)\;dx=\displaystyle\lim_{t \to -\infty}\int_{t}^{b} f(x)\;dx\)
     \((3)\;若\;f(x)\;在\;(-\infty,\infty)\;連續，則任意實數\;a\;，\int_{-\infty}^{\infty} f(x)\;dx=\int_{-\infty}^{a} f(x)\;dx+\int_{a}^{\infty} f(x)\;dx\)
     在以上任一狀況下，若極限存在，則稱瑕積分收斂 (convergent)，否則稱為發散 (divergent)


2. 冪次 \(p\) 的積分

     \((1)\;若\;p>1\;，則\;\int_{1}^{\infty} \frac{1}{x^p}\;dx\;收斂\)
     \((2)\;若\;p\le1\;，則\;\int_{1}^{\infty} \frac{1}{x^p}\;dx\;發散\)


3. 第二型瑕積分的定義

     \((1)\;若\;f(x)\;在\;[a, b)\;連續，則\;\int_{a}^{b} f(x)\;dx=\displaystyle\lim_{t \to b^-}\int_{a}^{t} f(x)\;dx\)
     \((2)\;若\;f(x)\;在\;(a,b]\;連續，則\;\int_{a}^{b} f(x)\;dx=\displaystyle\lim_{t \to a^+}\int_{t}^{b} f(x)\;dx\)
     \((3)\;令\;c\in(a,b)，若\;f(x)\;在\;[a,c)\cup(c,b] \;連續，且在\;x=c\;不連續，則\int_{a}^{b} f(x)\;dx=\int_{a}^{c} f(x)\;dx+\int_{c}^{b} f(x)\;dx\)
     在以上任一狀況下，若極限存在，則稱瑕積分收斂，否則稱為發散


4. 比較定理 (Comparison Theorem)

     假設 \(f\) 和 \(g\) 為連續函數， \(f(x)\ge g(x)\ge 0\;\)for\(\;x \ge a\;\)，則
     \((1)\;若\;\int_{a}^{\infty} f(x)\;dx\;收斂，則\;\int_{a}^{\infty} g(x)\;dx\;收斂\)
     \((2)\;若\;\int_{a}^{\infty} g(x)\;dx\;發散，則\;\int_{a}^{\infty} f(x)\;dx\;發散\)



Reference :

Essential Calculus, metric edition 2e, (2022) James Stewart, Daniel K. Clegg, Saleem Watson, Cengage Learning.
Paul’s Online Notes
The LibreTexts libraries Methmatics
Calculus Early Transcendentals:Differential & Multi-Variable Calculus for Social Sciences (2017). Petra Menz, Nicola Mulberry from Lyryx’ textbook