Integral Rules

\int 0 dx

c

\int 1 dx

x+c

\int a dx

ax+c

\int kf(x) dx

k\int f(x) dx

\int [f(x) \pm g(x)] dx

\int f(x) dx \pm \int g(x) dx

\int \sin x dx

-\cos(x)+c

\int x^n dx

\frac{x^{n+1}}{n+1}+c, n \neq -1

\int \cos x dx

\sin x+c

\int \sec^2 x dx

\tan x+c

\int \csc^2 x dx

-\cot x+c

\int \sec x \tan x dx

\sec x+c

\int \csc x \cot x dx

-\csc x+c

\int \frac{1}{x} dx

\ln |x|+c

\int a^x dx

\frac{a^x}{\ln a}+c

\int \tan x dx

\ln |\sec x|+c \text{ or } -\ln |\cos x|+c

\int \cot x dx

\ln |\sin x|+c \text{ or } -\ln |\csc x|+c

\int \sec x dx

\ln |\sec x+\tan x|+c

\int \csc x dx

\ln |\csc x-\cot x|+c

\int \frac{1}{\sqrt{1-x^2}} dx

\arcsin x+c

\int \frac{1}{1+x^2} dx

\arctan x+c

\int \frac{1}{|x|\sqrt{x^2-1}} dx

\operatorname{arcsec} x+c

\int e^x dx

e^x+c

\int \frac{1}{\sqrt{a^2 - x^2}} dx

\arcsin\left(\frac{x}{a}\right)+c

\int \frac{1}{a^2+x^2} dx

\frac{1}{a}\arctan\left(\frac{x}{a}\right)+c

\int \frac{1}{|x|\sqrt{x^2-a^2}} dx

\frac{1}{a}\operatorname{arcsec}|\frac{x}{a}|+c

\int \sin^2 x dx

\frac{x}{2}-\frac{\sin(2x)}{4}+c

\int \ln x dx

xln|x|-x+c