Calculus/Tables of Integrals

Rules

• $\int cf(x)\,dx=c\int f(x)\,dx$
• $\int f(x)+g(x)\,dx=\int f(x)\,dx+\int g(x)\,dx$
• $\int f(x)-g(x)\,dx=\int f(x)\,dx-\int g(x)\,dx$
• $\int u\,dv\ =uv-\int v\,du$

Powers

• $\int dx=x+C$
• $\int a\,dx=ax+C$
• $\int x^{n}\,dx={\frac {1}{n+1}}x^{{n+1}}+C\qquad {\mbox{ if }}n\neq -1$
• $\int {1 \over x}\,dx=\ln |x|+C$
• $\int {\frac {1}{ax+b}}\,dx={1 \over a}\ln |ax+b|+C\qquad {\mbox{ if }}a\neq 0$

Trigonometric Functions

Basic Trigonometric Functions

• $\int \sin {x}\,dx=-\cos {x}+C$
• $\int \cos {x}\,dx=\sin {x}+C$
• $\int \tan {x}\,dx=\ln \left|{\sec {x}}\right|+C$
• $\int \sin ^{2}{x}\,dx={\tfrac {1}{2}}x-{\tfrac {1}{4}}\sin {2x}+C$
• $\int \cos ^{2}{x}\,dx={\tfrac {1}{2}}x+{\tfrac {1}{4}}\sin {2x}+C$
• $\int \tan ^{2}{x}\,dx=\tan(x)-x+C$
• $\int \sin ^{n}x\,dx=-{\frac {\sin ^{{n-1}}x\cos x}{n}}+{\frac {n-1}{n}}\int \sin ^{{n-2}}x\,dx+C\qquad {\mbox{(for }}n>0{\mbox{)}}$

• $\int \cos ^{n}x\,dx=-{\frac {\cos ^{{n-1}}x\sin x}{n}}+{\frac {n-1}{n}}\int \cos ^{{n-2}}x\,dx+C\qquad {\mbox{(for }}n>0{\mbox{)}}\,\!$

• $\int \tan ^{n}x\,dx={\frac {1}{(n-1)}}\tan ^{{n-1}}x-\int \tan ^{{n-2}}x\,dx+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}$

Reciprocal Trigonometric Functions

• $\int \sec {x}\,dx=\ln \left|{\sec {x}}+\tan x\right|+C=\ln \left|\tan {\left({\frac 12}x+{\frac 14}\pi \right)}\right|+C$
• $\int \csc {x}\,dx=-\ln \left|{\csc x+\cot x}\right|+C=\ln \left|\tan \left({\frac 12}x\right)\right|+C$
• $\int \cot {x}\,dx=\ln \left|{\sin {x}}\right|+C$

• $\int \sec ^{2}kx\,dx={\frac 1k}\tan {kx}+C$
• $\int \csc ^{2}kx\,dx=-{\frac 1k}\cot kx+C$
• $\int \cot ^{2}kx\,dx=-x-{\frac 1k}\cot kx+C$
• $\int \sec {x}\tan {x}\,dx=\sec x+C$
• $\int \sec x\csc x\,dx=\ln \left|\tan x\right|+C$

• $\int \sec ^{n}{x}\,dx={\frac {\sec ^{{n-1}}{x}\sin {x}}{n-1}}+{\frac {n-2}{n-1}}\int \sec ^{{n-2}}{x}\,dx+C\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}$

• $\int \csc ^{n}{x}\,dx=-{\frac {\csc ^{{n-1}}{x}\cos {x}}{n-1}}+{\frac {n-2}{n-1}}\int \csc ^{{n-2}}{x}\,dx+C\qquad {\mbox{ (for }}n\neq 1{\mbox{)}}$

• $\int \cot ^{n}x\,dx=-{\frac {1}{n-1}}\cot ^{{n-1}}x-\int \cot ^{{n-2}}x\,dx+C\qquad {\mbox{(for }}n\neq 1{\mbox{)}}$

Inverse Trigonometric Functions

• $\int {1 \over {\sqrt {1-x^{2}}}}\,dx={\mbox{arcsin}}(x)+C$
• $\int {1 \over {\sqrt {a^{2}-x^{2}}}}\,dx={\mbox{arcsin}}(x/a)+C\qquad {\mbox{ if }}a\neq 0$
• $\int {1 \over 1+x^{2}}\,dx={\mbox{arctan}}(x)+C$
• $\int {1 \over a^{2}+x^{2}}\,dx={1 \over a}{\mbox{arctan}}(x/a)+C\qquad {\mbox{ if }}a\neq 0$

Exponential and Logarithmic Functions

• $\int e^{x}\,dx=e^{x}+C$
• $\int e^{{ax}}\,dx={1 \over a}e^{{ax}}+C\qquad {\mbox{ if }}a\neq 0$
• $\int a^{x}\,dx={1 \over \ln a}a^{x}+C\qquad {\mbox{ if }}a>0,a\neq 1$
• $\int \ln x\,dx=x\ln x-x+C$

Inverse Trigonometric Functions

• $\int {\mbox{arcsin}}(x)\,dx=x\,{\mbox{arcsin}}(x)+{\sqrt {1-x^{2}}}+C$
• $\int {\mbox{arccos}}(x)\,dx=x\,{\mbox{arccos}}(x)-{\sqrt {1-x^{2}}}+C$
• $\int {\mbox{arctan}}(x)\,dx=x\,{\mbox{arctan}}(x)-{1 \over 2}\ln(1+x^{2})+C$