Calculus/Tables of Integrals

Rules

• $\int c\cdot f(x)\mathrm{d}x=c\cdot \int f(x)\mathrm{d}x$
• $\int (f(x)\pm g(x))\mathrm{d}x=\int f(x)\mathrm{d}x\pm \int g(x)\mathrm{d}x$
• $\int f(g(x))g^{\prime }(x)\mathrm{d}x=\int f(u)\mathrm{d}u=F(u)+C=F(g(x))+C$ where $F^{\prime }=f$
• $\int udv=uv-\int vdu$

Powers

• $\int \mathrm{d}x=x+C$
• $\int a\mathrm{d}x=ax+C$
• $\int x^n\mathrm{d}x=\frac{x^{n+1}}{n+1}+C(\text{for }n\ne -1)$
• $\int \frac{\mathrm{d}x}{x}=\ln |x|+C$
• $\int \frac{\mathrm{d}x}{ax+b}=\frac{\ln |ax+b|}{a}+C(\text{for }a\ne 0)$

Trigonometric Functions

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

• $\int \sec (x)\mathrm{d}x=\ln |\sec (x)+\tan (x)|+C=\ln |\tan (\frac{x}{2}+\frac{\pi }{4})|+C=2\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(\tan \left(\frac{x}{2}\right))+C$
• $\int \csc (x)\mathrm{d}x=-\ln |\csc (x)+\cot (x)|+C=\ln |\tan \left(\frac{x}{2}\right)|+C$
• $\int \cot (x)\mathrm{d}x=\ln |\sin (x)|+C$

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

Reduction formulae

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

Explicit forms

• $\int {\sin }^n(x)\mathrm{d}x=-\cos (x{)}_2{F}_1(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};{\cos }^2(x))+C$
• $\int {\cos }^n(x)\mathrm{d}x=-\frac{1}{n+1}\mathrm{s}\mathrm{g}\mathrm{n}(\sin (x)){\cos }^{n+1}(x{)}_2{F}_1(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};{\cos }^2(x))+C(\text{for }n\ne -1)$
• $\int {\tan }^n(x)\mathrm{d}x=\frac{1}{n+1}{\tan }^{n+1}(x{)}_2{F}_1(1,\frac{n+1}{2};\frac{n+3}{2};-{\tan }^2(x))+C(\text{for }n\ne -1)$
• $\int {\csc }^n(x)\mathrm{d}x=-\cos (x{)}_2{F}_1(\frac{1}{2},\frac{n+1}{2};\frac{3}{2};{\cos }^2(x))+C$
• $\int {\sec }^n(x)\mathrm{d}x=\sin (x{)}_2{F}_1(\frac{1}{2},\frac{n+1}{2};\frac{3}{2};{\sin }^2(x))+C$
• $\int {\cot }^n(x)\mathrm{d}x=-\frac{1}{n+1}{\cot }^{n+1}(x{)}_2{F}_1(1,\frac{n+1}{2};\frac{n+3}{2};-{\cot }^2(x))+C(\text{for }n\ne -1)$

Where ${}_2{F}_1$ is the hypergeometric function and $\mathrm{s}\mathrm{g}\mathrm{n}$ is the sign function.

Inverse Trigonometric Functions

• $\int \frac{\mathrm{d}x}{\sqrt{1-x^2}}=\arcsin (x)+C$
• $\int \frac{\mathrm{d}x}{\sqrt{a^2-x^2}}=\arcsin \left(\frac{x}{a}\right)+C(\text{for }a\ne 0)$
• $\int \frac{\mathrm{d}x}{1+x^2}=\arctan (x)+C$
• $\int \frac{\mathrm{d}x}{a^2+x^2}=\frac{\arctan \left(\frac{x}{a}\right)}{a}+C(\text{for }a\ne 0)$

Exponential and Logarithmic Functions

• $\int e^x\mathrm{d}x=e^x+C$
• $\int e^{ax}\mathrm{d}x=\frac{e^{ax}}{a}+C(\text{for }a\ne 0)$
• $\int a^x\mathrm{d}x=\frac{a^x}{\ln (a)}+C(\text{for }a>0,a\ne 1)$
• $\int \ln (x)\mathrm{d}x=x\ln (x)-x+C$
• $\int e^x\sin (x)\mathrm{d}x=\frac{e^x}{2}(\sin (x)-\cos (x))+C$
• $\int e^x\cos (x)\mathrm{d}x=\frac{e^x}{2}(\sin (x)+\cos (x))+C$

Reduction formulae

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

Inverse Trigonometric Functions

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

Hyperbolic functions

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

Reciprocals

• $\int \mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}(x)\mathrm{d}x=i\int \csc (ix)\mathrm{d}x=\log |-i\tan \left(\frac{ix}{2}\right)|+C=\log |\tanh \left(\frac{x}{2}\right)|+C$
• $\int \mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x)\mathrm{d}x=\int \sec (ix)\mathrm{d}x=2\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(-i\tan \left(\frac{x}{2}i\right))+C=2\arctan (\tanh \left(\frac{x}{2}\right))+C$
• $\int \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(x)\mathrm{d}x=i\int \cot (ix)\mathrm{d}x=\log |-i\sin (ix)|+C=\log |\sinh (x)|+C$

Inverses

• $\int \mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(x)\mathrm{d}x=x\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(x)-\sqrt{x^2+1}+C$
• $\int \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(x)\mathrm{d}x=x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}(x)-\sqrt{x^2-1}+C$
• $\int \mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(x)\mathrm{d}x=x\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(x)+\frac{1}{2}\ln (1-x^2)+C$
• $\int \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}(x)\mathrm{d}x=x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}(x)+|\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(x)|+C$
• $\int \mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x)\mathrm{d}x=x\mathrm{a}\mathrm{r}\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}(x)+\arcsin (x)+C$
• $\int \mathrm{a}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(x)\mathrm{d}x=x\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(x)+\frac{1}{2}\ln (x^2-1)+C$

Misc

• $\int |f(x)|\mathrm{d}x=\mathrm{s}\mathrm{g}\mathrm{n}(f(x))\int f(x)\mathrm{d}x$, where $\mathrm{s}\mathrm{g}\mathrm{n}$ is the sign function.

Definite integrals

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