Calculus/Tables of Integrals

Rules

 * $$\int c\cdot f(x)\mathrm{d}x=c\cdot\int f(x)\mathrm{d}x$$
 * $$\int\big(f(x)\pm g(x)\big)\mathrm{d}x=\int f(x)\mathrm{d}x\pm\int g(x)\mathrm{d}x$$
 * $$\int f(g(x)) g'(x) \mathrm{d}x = \int f(u) \mathrm{d}u = F(u) + C = F(g(x)) + C$$ where $$ F' = f$$
 * $$\int u\,dv=uv-\int v\,du$$

Powers

 * $$\int \mathrm{d}x=x+C$$
 * $$\int a\,\mathrm{d}x=ax+C$$
 * $$\int x^n\mathrm{d}x=\frac{x^{n+1}}{n+1}+C\qquad(\text{for }n\ne-1)$$
 * $$\int\frac{\mathrm{d}x}{x}=\ln|x|+C$$
 * $$\int\frac{\mathrm{d}x}{ax+b}=\frac{\ln|ax+b|}{a}+C\qquad(\text{for }a\ne0)$$

Basic Trigonometric Functions

 * $$\int\sin(x)\mathrm{d}x=-\cos(x)+C$$
 * $$\int\cos(x)\mathrm{d}x=\sin(x)+C$$
 * $$\int\tan(x)\mathrm{d}x=\ln|\sec(x)|+C$$
 * $$\int\sin^2(x)\mathrm{d}x=\int\frac{1-\cos(2x)}{2}\mathrm{d}x=\frac{x}{2}-\frac{\sin(2x)}{4}+C$$
 * $$\int\cos^2(x)\mathrm{d}x=\int\frac{1+\cos(2x)}{2}\mathrm{d}x=\frac{x}{2}+\frac{\sin(2x)}{4}+C$$
 * $$\int\tan^2(x)\mathrm{d}x=\tan(x)-x+C$$

Reciprocal Trigonometric Functions

 * $$\int\sec(x)\mathrm{d}x=\ln\Big|\sec(x)+\tan(x)\Big|+C=\ln\left|\tan\left(\frac{x}{2}+\frac{\pi}{4}\right)\right|+C=2\mathrm{artanh}\left(\tan\left(\frac{x}{2}\right)\right)+C$$
 * $$\int\csc(x)\mathrm{d}x=-\ln\Big|\csc(x)+\cot(x)\Big|+C=\ln\left|\tan\left(\frac{x}{2}\right)\right|+C$$
 * $$\int\cot(x)\mathrm{d}x=\ln|\sin(x)|+C$$


 * $$\int\sec^2(ax)\mathrm{d}x=\frac{\tan(ax)}{a}+C$$
 * $$\int\csc^2(ax)\mathrm{d}x=-\frac{\cot(ax)}{a}+C$$
 * $$\int\cot^2(ax)\mathrm{d}x=-x-\frac{\cot(ax)}{a}+C$$
 * $$\int\sec(x)\tan(x)\mathrm{d}x=\sec(x)+C$$
 * $$\int\sec(x)\csc(x)\mathrm{d}x=\ln|\tan(x)|+C$$

Reduction formulae

 * $$\int\sin^n(x)\mathrm{d}x=-\frac{\sin^{n-1}(x)\cos(x)}{n}+\frac{n-1}{n}\int\sin^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n>0)$$
 * $$\int\cos^n(x)\mathrm{d}x=-\frac{\cos^{n-1}(x)\sin(x)}{n}+\frac{n-1}{n}\int\cos^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n>0)$$
 * $$\int\tan^n(x)\mathrm{d}x=\frac{\tan^{n-1}(x)}{(n-1)}-\int\tan^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n\ne1)$$
 * $$\int\sec^n(x)\mathrm{d}x=\frac{\sec^{n-1}(x)\sin(x)}{n-1}+\frac{n-2}{n-1}\int\sec^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n\ne1)$$
 * $$\int\csc^n(x)\mathrm{d}x=-\frac{\csc^{n-1}(x)\cos(x)}{n-1}+\frac{n-2}{n-1}\int\csc^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n\ne1)$$
 * $$\int\cot^n(x)\mathrm{d}x=-\frac{\cot^{n-1}(x)}{n-1}-\int\cot^{n-2}(x)\mathrm{d}x+C\qquad(\text{for }n\ne1)$$
 * $$a^2\int x^n\sin(ax)\mathrm{d}x = nx^{n-1}\sin(ax)-ax^n\cos(ax)-n(n-1)\int x^{n-2}\sin(ax)\mathrm{d}x$$
 * $$a^2\int x^n\cos(ax)\mathrm{d}x = ax^n\sin(ax)+nx^{n-1}\cos(ax)-n(n-1)\int x^{n-2}\cos(ax)\mathrm{d}x$$

Explicit forms

 * $$\int \sin^n(x)\mathrm{d}x = -\cos(x)_2F_1\left(\frac{1}{2}, \frac{1-n}{2}; \frac{3}{2}; \cos^2(x)\right) + C$$
 * $$\int \cos^n(x)\mathrm{d}x = -\frac{1}{n+1}\mathrm{sgn}(\sin(x))\cos^{n+1}(x)_2F_1\left(\frac{1}{2}, \frac{n+1}{2}; \frac{n+3}{2}; \cos^2(x)\right) + C\qquad(\text{for }n\ne-1)$$
 * $$\int \tan^n(x)\mathrm{d}x = \frac{1}{n+1}\tan^{n+1}(x)_2F_1\left(1, \frac{n+1}{2}; \frac{n+3}{2}; -\tan^2(x)\right) + C\qquad(\text{for }n\ne-1)$$
 * $$\int \csc^n(x)\mathrm{d}x = -\cos(x)_2F_1\left(\frac{1}{2}, \frac{n+1}{2}; \frac{3}{2}; \cos^2(x)\right) + C$$
 * $$\int \sec^n(x)\mathrm{d}x = \sin(x)_2F_1\left(\frac{1}{2}, \frac{n+1}{2}; \frac{3}{2}; \sin^2(x)\right) + C$$
 * $$\int \cot^n(x)\mathrm{d}x = -\frac{1}{n+1}\cot^{n+1}(x)_2F_1\left(1, \frac{n+1}{2}; \frac{n+3}{2}; -\cot^2(x)\right) + C\qquad(\text{for }n\ne-1)$$

Where $${}_2F_1$$ is the hypergeometric function and $$\mathrm{sgn}$$ is the sign function.

Inverse Trigonometric Functions

 * $$\int\frac{\mathrm{d}x}{\sqrt{1-x^2}}=\arcsin(x)+C$$
 * $$\int\frac{\mathrm{d}x}{\sqrt{a^2-x^2}}=\arcsin\left(\tfrac{x}{a}\right)+C\qquad(\text{for }a\ne0)$$
 * $$\int\frac{\mathrm{d}x}{1+x^2}=\arctan(x)+C$$
 * $$\int\frac{\mathrm{d}x}{a^2+x^2}=\frac{\arctan\left(\tfrac{x}{a}\right)}{a}+C\qquad(\text{for }a\ne0)$$

Exponential and Logarithmic Functions

 * $$\int e^x\mathrm{d}x=e^x+C$$
 * $$\int e^{ax}\mathrm{d}x=\frac{e^{ax}}{a}+C\qquad(\text{for }a\ne0)$$
 * $$\int a^x\mathrm{d}x=\frac{a^x}{\ln(a)}+C\qquad(\text{for }a>0,a\ne1)$$
 * $$\int\ln(x)\mathrm{d}x=x\ln(x)-x+C$$
 * $$\int e^{x}\sin(x)\mathrm{d}x = \frac{e^{x}}{2}(\sin(x) - \cos(x)) + C$$
 * $$\int e^{x}\cos(x)\mathrm{d}x = \frac{e^{x}}{2}(\sin(x) + \cos(x)) + C$$

Reduction formulae

 * $$\int x^ne^{ax}\mathrm{d}x = \frac{1}{a}x^ne^{ax} - \frac{n}{a}\int x^{n-1}e^{ax}\mathrm{d}x$$

Inverse Trigonometric Functions

 * $$\int\arcsin(x)\mathrm{d}x=x\arcsin(x)+\sqrt{1-x^2}+C$$
 * $$\int\arccos(x)\mathrm{d}x=x\arccos(x)-\sqrt{1-x^2}+C$$
 * $$\int\arctan(x)\mathrm{d}x=x\arctan(x)-\frac{1}{2}\ln|1+x^2|+C$$
 * $$\int\arccsc(x)\mathrm{d}x=x\arccsc(x)+\ln\left|x + x\sqrt{1-\frac{1}{x^2}}\right| + C$$
 * $$\int\arcsec(x)\mathrm{d}x=x\arcsec(x)-\ln\left|x + x\sqrt{1-\frac{1}{x^2}}\right| + C$$
 * $$\int\arccot(x)\mathrm{d}x=x\arccot(x)+\frac{1}{2}\ln|1+x^2| + C$$

Hyperbolic functions

 * $$\int \sinh(x)\mathrm{d}x = -i\int \sin(ix) \mathrm{d}x = \cos(ix) + C = \cosh(x) + C$$
 * $$\int \cosh(x)\mathrm{d}x = \int \cos(ix) \mathrm{d}x = -i\sin(ix) + C = \sinh(x) + C$$
 * $$\int \tanh(x)\mathrm{d}x = -i\int \tan(ix) \mathrm{d}x = \log\left|\cos(ix)\right| + C = \log\left|\cosh(x)\right| + C$$

Reciprocals
2\arctan\left(\tanh\left(\frac{x}{2}\right)\right) + C$$
 * $$\int \mathrm{csch}(x)\mathrm{d}x = i\int \csc(ix) \mathrm{d}x = \log\left|-i\tan\left(\frac{ix}{2}\right)\right| + C = \log\left|\tanh\left(\frac{x}{2}\right)\right| + C$$
 * $$\int \mathrm{sech}(x)\mathrm{d}x = \int \sec(ix) \mathrm{d}x = 2\mathrm{artanh}\left(-i\tan\left(\frac{x}{2}i\right)\right) + C =
 * $$\int \mathrm{coth}(x) \mathrm{d}x = i\int \cot(ix) \mathrm{d}x = \log\left|-i\sin(ix)\right| + C = \log\left|\sinh(x)\right| + C$$

Inverses

 * $$\int \mathrm{arsinh}(x)\mathrm{d}x = x\mathrm{arsinh}(x) - \sqrt{x^2 + 1} + C$$
 * $$\int \mathrm{arcosh}(x)\mathrm{d}x = x\mathrm{arcosh}(x) - \sqrt{x^2 - 1} + C$$
 * $$\int \mathrm{artanh}(x)\mathrm{d}x = x\mathrm{artanh}(x) + \frac{1}{2}\ln(1-x^2) + C$$
 * $$\int \mathrm{arcsch}(x)\mathrm{d}x = x\mathrm{arcsch}(x) + |\mathrm{arsinh}(x)| + C$$
 * $$\int \mathrm{arsech}(x)\mathrm{d}x = x\mathrm{arsech}(x) + \arcsin(x) + C$$
 * $$\int \mathrm{artanh}(x)\mathrm{d}x = x\mathrm{arcoth}(x) + \frac{1}{2}\ln(x^2 - 1) + C$$

Misc

 * $$\int |f(x)|\mathrm{d}x = \mathrm{sgn}(f(x))\int f(x)\mathrm{d}x$$, where $$\mathrm{sgn}$$ is the sign function.

Definite integrals

 * $$\int_{[0,1]^n} \frac{\prod_{i=1}^n\mathrm{d}x_i}{1-\prod_{i=1}^n x_i} = \zeta(n) \text{ for all integers } n > 1$$, where $$\zeta$$ is the Riemann zeta function.
 * $$\int_{-\infty}^{\infty} e^{-x^2} \mathrm{d}x = \sqrt{\pi}$$
 * $$\int_0^1 t^{u-1}(1-t)^{v-1}\mathrm{d}t = \beta(u,v) = \frac{\Gamma(u)\Gamma(v)}{\Gamma(u+v)}$$, where $$\Gamma$$ is the gamma function.
 * $$\int_0^\infty t^{s-1}e^{-t}\mathrm{d}t = \Gamma(s)$$
 * $$\int_0^{2\pi} e^{u\cos\theta}\mathrm{d}\theta = 2\pi I_0(u)$$, where $$I_0$$ is the modified Bessel function of the first kind.
 * $$\int_0^\infty \frac{\sin(x)}{x} \mathrm{d}x = \frac{\pi}{2}$$