Calculus/Tables of Integrals

Rules

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

Powers

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

Trigonometric Functions

Basic Trigonometric Functions

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

Reciprocal Trigonometric Functions

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

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

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

Inverse Trigonometric Functions

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

Exponential and Logarithmic Functions

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

Inverse Trigonometric Functions

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