DEFINITE INTEGRALS THAT CONTAIN TRIGONOMETRIC FUNCTIONS

Note that all the constant are positive.

1.

$\displaystyle\int_{0}^{\pi}\sin mx \sin nx dx=\left\{ \begin{array}{ll} \displa... ...in} &\mbox{if $m$\space and $n$\space integers and}\;\; m=n \end{array}\right.$

2.

$\displaystyle\int_{0}^{\pi} \cos mx \cos nx dx=\left\{ \begin{array}{ll} \displ... ...in} &\mbox{if $m$\space and $n$\space integers and}\;\; m=n \end{array}\right.$

3.

$\displaystyle\int_{0}^{\pi}\sin mx \cos nx dx=\left\{ \begin{array}{ll} \displa... ... $m$\space and $n$\space integers and $m+n$\space even} \\ \end{array}\right.$

4.

$\displaystyle\int_{0}^{\pi/2} \sin^2 x dx=\int_{0}^{\pi/2}\cos^2 dx=\displaystyle \frac{\pi}{4}$

5.

$\displaystyle\int_{0}^{\pi/2}\sin^{2m} x dx=\int_{0}^{\pi/2}\cos^{2m} x dx=\dis... ...\cdot 4\cdot 6\cdot \cdot\cdot\cdot 2m}\left(\displaystyle \frac{\pi}{2}\right)$ ,
m=1,2,...

6.

$\displaystyle\int_{0}^{\pi/2}\sin^{2m+1}x dx=\int_{0}^{\pi/2}\cos^{2m+1}x dx=\d... ...c{2\cdot 4\cdot 6\cdot\cdot\cdot 2m}{1\cdot 3\cdot 5\cdot \cdot\cdot\cdot 2m+1}$ ,
m=1,2,...

7.

$\displaystyle\int_{0}^{\pi/2}\sin^{2p-1}x \cos^{2q-1}x dx=\displaystyle \frac{\Gamma(p)\Gamma(q)}{2\Gamma(p+q)}$

8.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin px}{x}dx=\left\{ \begin{... ...n} p>0 \\ 0&\hspace{.3in} p=0 \\ -\pi/2&\hspace{.3in} p<0 \end{array}\right.$

9.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin px\cos qx}{x}dx=\left\{ ... ...0\\ \pi/2&\hspace{.3in} 0<p<q\\ \pi/4&\hspace{.3in} p=q>0 \end{array}\right.$

10.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin px \sin qx}{x^2}dx=\left... ...\hspace{.3in} 0<p\leq q \\ \pi q/2&\hspace{.3in} p\geq q>0 \end{array}\right.$

11.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin^2 px}{x^2}dx=\displaystyle \frac{\pi p}{2}$

12.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{1-\cos px}{x^2}dx=\displaystyle \frac{\pi p}{2}$

13.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\cos px-\cos qx}{x}dx=\ln\displaystyle \frac{q}{p}$

14.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\cos px-\cos qx}{x^2}dx=\displaystyle \frac{\pi(q-p)}{2}$

15.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\cos mx}{x^2+a^2}dx=\displaystyle \frac{\pi}{2a}e^{-ma}$

16.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{x\sin mx}{x^2+a^2}dx=\displaystyle \frac{\pi}{2}e^{-ma}$

17.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin mx}{x(x^2+a^2)}dx=\displaystyle \frac{\pi}{2a^2}(1-e^{-ma})$

18.

$\displaystyle\int_{0}^{2\pi}\displaystyle \frac{dx}{a+b\sin x}=\displaystyle \frac{2\pi}{\displaystyle \sqrt{a^2-b^2}}$

19.

$\displaystyle\int_{0}^{2\pi}\displaystyle \frac{dx}{a+b\cos x}=\displaystyle \frac{2\pi}{\displaystyle \sqrt{a^2-b^2}}$

20.

$\displaystyle\int_{0}^{\pi/2}\displaystyle \frac{dx}{a+b\cos x}=\displaystyle \frac{cos^{-1}(b/a)}{\displaystyle \sqrt{a^2-b^2}}$

21.

$\displaystyle\int_{0}^{2\pi}\displaystyle \frac{dx}{(a+b\sin x)^2}=\int_{0}^{2\... ...playstyle \frac{dx}{(a+b\cos x)^2}=\displaystyle \frac{2\pi a}{(a^2-b^2)^{3/2}}$

22.

$\displaystyle\int_{0}^{2\pi}\displaystyle \frac{dx}{1-2a\cos x+a^2}=\displaystyle \frac{2\pi}{1-a^2},\hspace{.2in}0<a<1$

23.

$\displaystyle\int_{0}^{\pi}\displaystyle \frac{x \sin x dx}{1-2a\cos x +a^2}=\l... ...\mid a\mid <1\\ \pi\ln(1+1/a) &\hspace{.3in} \mid a\mid >1 \end{array}\right.$

24.

$\displaystyle\int_{0}^{\pi}\displaystyle \frac{\cos mx dx}{1-2a\cos x+a^2}=\displaystyle \frac{\pi a^m}{1-a^2},\hspace{.2in}a^2<1$ ,
m=0,1,2,...

25.

$\displaystyle\int_{0}^{\infty}\sin ax^2 dx=\int_{0}^{\infty}\cos ax^2 dx=\displaystyle \frac{1}{2}\displaystyle \sqrt{\displaystyle \frac{\pi}{2a}}$

26.

$\displaystyle\int_{0}^{\infty}\sin ax^n dx=\displaystyle \frac{1}{na^{1/n}}\Gamma(1/n)\sin\displaystyle \frac{\pi}{2n},\hspace{.2in}n>1$

27.

$\displaystyle\int_{0}^{\infty}\cos ax^n dx=\displaystyle \frac{1}{na^{1/n}}\Gamma(1/n)\cos\displaystyle \frac{\pi}{2n},\hspace{.2in}n>1$

28.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin x}{\displaystyle \sqrt{x... ...s x}{\displaystyle \sqrt{x}}dx=\displaystyle \sqrt{\displaystyle \frac{\pi}{2}}$

29.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin x}{x^p}dx=\displaystyle \frac{\pi}{2\Gamma (p)\sin(p\pi/2)},\hspace{.2in}0<p<1$

30.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\cos x}{x^p}dx=\displaystyle \frac{\pi}{2\Gamma (p)\cos(p\pi/2)},\hspace{.2in}0<p<1$

31.

$\displaystyle\int_{0}^{\infty}\sin ax^2 \cos 2bx dx=\displaystyle \frac{1}{2}\d... ...}}\left( \cos\displaystyle \frac{b^2}{a}-\sin\displaystyle \frac{b^2}{a}\right)$

32.

$\displaystyle\int_{0}^{\infty}\cos ax^2\cos 2bxdx=\displaystyle \frac{1}{2}\dis... ...}}\left( \cos\displaystyle \frac{b^2}{a}+\sin\displaystyle \frac{b^2}{a}\right)$

33.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin^3 x}{x^3}dx=\displaystyle \frac{3\pi}{8}$

34.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\sin^4 x}{x^4}dx=\displaystyle \frac{\pi}{3}$

35.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\tan x}{x}dx=\displaystyle \frac{\pi}{2}$

36.

$\displaystyle\int_{0}^{\pi/2}\displaystyle \frac{dx}{1+\tan^m x}=\displaystyle \frac{\pi}{4}$

37.

$\displaystyle\int_{0}^{\pi/2}\displaystyle \frac{x}{\sin x}dx=2\left\{ \display... ...displaystyle \frac{1}{5^2}-\displaystyle \frac{1}{7^2}+\cdot\cdot\cdot \right\}$

38.

$\displaystyle\int_{0}^{1}\displaystyle \frac{\tan^{-1}x}{x}dx=\displaystyle \fr... ...}{3^2}+\displaystyle \frac{1}{5^2}-\displaystyle \frac{1}{7^2}+ \cdot\cdot\cdot$

39.

$\displaystyle\int_{0}^{1}\displaystyle \frac{\sin^{-1}x}{x}dx=\displaystyle \frac{\pi}{2}\ln2$

40.

$\displaystyle\int_{0}^{1}\displaystyle \frac{1-\cos x}{x}dx-\int_{1}^{\infty}\displaystyle \frac{\cos x}{x}dx=\gamma$

where the constant $\gamma$ is the Euler's constant.

41.

$\displaystyle\int_{0}^{\infty}\left( \displaystyle \frac{1}{1+x^2}-\cos x\right)\displaystyle \frac{dx}{x}=\gamma$

where the constant $\gamma$ is the Euler's constant.

42.

$\displaystyle\int_{0}^{\infty}\displaystyle \frac{\tan^{-1}px-\tan^{-1}qx}{x}dx=\displaystyle \frac{\pi}{2}\ln\displaystyle \frac{p}{q}$