Calculus/Tables of Integrals

Rules

• $\int c\cdot f(x)dx=c\cdot \int f(x)dx$
• $\int {\big (}f(x)\pm g(x){\big )}dx=\int f(x)dx\pm \int g(x)dx$
• $\int u\,dv=uv-\int v\,du$

Powers

• $\int dx=x+C$
• $\int a\,dx=ax+C$
• $\int x^{n}dx={\frac {x^{n+1}}{n+1}}+C\qquad ({\text{for }}n\neq -1)$
• $\int {\frac {dx}{x}}=\ln |x|+C$
• $\int {\frac {dx}{ax+b}}={\frac {\ln |ax+b|}{a}}+C\qquad ({\text{for }}a\neq 0)$

Trigonometric Functions

Basic Trigonometric Functions

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

Reciprocal Trigonometric Functions

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

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

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

Inverse Trigonometric Functions

• $\int {\frac {dx}{\sqrt {1-x^{2}}}}=\arcsin(x)+C$
• $\int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\arcsin \left({\tfrac {x}{a}}\right)+C\qquad ({\text{for }}a\neq 0)$
• $\int {\frac {dx}{1+x^{2}}}=\arctan(x)+C$
• $\int {\frac {dx}{a^{2}+x^{2}}}={\frac {\arctan \left({\tfrac {x}{a}}\right)}{a}}+C\qquad ({\text{for }}a\neq 0)$

Exponential and Logarithmic Functions

• $\int e^{x}dx=e^{x}+C$
• $\int e^{ax}dx={\frac {e^{ax}}{a}}+C\qquad ({\text{for }}a\neq 0)$
• $\int a^{x}dx={\frac {a^{x}}{\ln(a)}}+C\qquad ({\text{for }}a>0,a\neq 1)$
• $\int \ln(x)dx=x\ln(x)-x+C$

Inverse Trigonometric Functions

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