Integration Rules

Basic

• $\int 0=C$
• $\int kdx=kx+C$
• $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$
• $\int e^{x}dx=e^x+C$
• $\int a^{x}dx=\frac{a^x}{\ln a}+C$
• $\int \frac{1}{x}=\ln |x|+C$

Trig

• $\int \sin xdx=-\cos x+C$
• $\int \cos xdx=\sin x+C$
• $\int {\sec }^{2}xdx=\tan x+C$
• $\int {\csc }^{2}xdx=-\cot x+C$
• $\int \sec x\tan xdx=\sec x+C$
• $\int \csc x\cot xdx=-\csc x+C$
• $\int \tan xdx=\ln |\sec x|+C=-\ln |\cos x|+C$
• $\int \cot xdx=\ln |\sin x|+C=-\ln |\csc u|+C$
• $\int \sec xdx=\ln |\sec x+\tan x|+C$
• $\int \csc xdx=-\ln |\csc x+\cot x|+C$

Inverse Trig

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