Calculus/Tables of Integrals

Rules

• $\int cf(x)dx=c\int f(x)dx$
• $\int f(x)+g(x)dx=\int f(x)dx+\int g(x)dx$
• $\int f(x)-g(x)dx=\int f(x)dx-\int g(x)dx$
• $\int udv=uv-\int vdu$

Powers

• $\int dx=x+C$
• $\int adx=ax+C$
• $\int x^{n}dx=\frac{1}{n+1}{x}^{n+1}+C\text{ if }n\ne -1$
• $\int x^{-n}dx=\frac{1}{-n+1}{x}^{-n+1}+C\text{ if }n\ne 1$
• $\int \frac{1}{x}dx=\ln |x|+C$
• $\int \frac{1}{ax+b}dx=\frac{1}{a}\ln |ax+b|+C\text{ if }a\ne 0$

Trigonometric Functions

Basic Trigonometric Functions

• $\int \sin xdx=-\cos x+C$
• $\int \cos xdx=\sin x+C$
• $\int \tan xdx=\ln \left|\sec x\right|+C$
• $\int {\sin }^{2}xdx=\frac{1}{2}x-\frac{1}{4}\sin 2x+C$
• $\int {\cos }^{2}xdx=\frac{1}{2}x+\frac{1}{4}\sin 2x+C$
• $\int {\tan }^{2}xdx=\tan (x)-x+C$

Reciprocal Trigonometric Functions

• $\int \sec xdx=\ln |\sec x+\tan x|+C=\ln |\tan (\frac{1}{2}x+\frac{1}{4}\pi )|+C$
• $\int \csc xdx=-\ln \left|\csc x+\cot x\right|+C=\ln |\tan \left(\frac{1}{2}x\right)|+C$
• $\int \cot xdx=\ln \left|\sin x\right|+C$

• $\int {\sec }^{2}kxdx=\frac{1}{k}\tan kx+C$
• $\int {\csc }^{2}kxdx=-\frac{1}{k}\cot kx+C$
• $\int {\cot }^{2}kxdx=-x-\frac{1}{k}\cot kx+C$

• $\int \sec x\tan xdx=\sec x+C$
• $\int {\csc }^{n}kx\cot kxdx=-\frac{1}{ka}{\csc }^{n}ax+C$
• $\int \sec x\csc xdx=\ln |\tan x|+C$

Inverse Trigonometric Functions

• $\int \frac{1}{\sqrt{1-x^2}}dx=\text{arcsin}(x)+C$
• $\int \frac{1}{\sqrt{a^2-x^2}}dx=\frac{1}{a}\text{arcsin}(x/a)+C\text{ if }a\ne 0$
• $\int \frac{1}{1+x^2}dx=\text{arctan}(x)+C$
• $\int \frac{1}{a^2+x^2}dx=\frac{1}{a}\text{arctan}(x/a)+C\text{ if }a\ne 0$

Exponential and Logarithmic Functions

• $\int e^{x}dx=e^x+C$
• $\int e^{ax}dx=\frac{1}{a}{e}^{ax}+C\text{ if }a\ne 0$
• $\int a^{x}dx=\frac{1}{\ln a}{a}^x+C\text{ if }a>0,a\ne 1$
• $\int \ln xdx=x\ln x-x+C$

Inverse Trigonometric Functions

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

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