Table of integrals

The following is a table of indefinite integrals, or antiderivatives. C is the constant of integration.

Standard functions

• $\int a\,dx=ax+C$
• $\int x^ndx=\frac{x^{n+1}}{n+1}+C\ ,\ n\ne-1$
• $\int f(x)g(x)dx=f(x)\int g(x)dx-\int f'(x)\left(\int g(x)dx\right)dx+C$(integration by parts)

Logarithmic and exponential functions

• $\int a^xdx=\dfrac{a^x}{\ln(a)}+C$
• $\int e^x dx = e^x + C$
• $\int\dfrac{dx}{x}=\ln\big(|x|\big)+C$
• $\int \ln(x) dx = x \ln (x) - x + C$
• $\int\log_a(x)dx=x\log_a(x)-\dfrac{x}{\ln(a)}+C$

Trigonometric functions

• $\int \sin(x) dx = -\cos (x) + C$
• $\int \cos(x) dx = \sin (x) + C$
• $\int\tan(x)dx=-\ln\big(|\cos(x)|\big)+C=\ln\big(|\sec(x)|\big)+C$
• $\int\csc(x)dx=\ln\left(\left|\tan\left(\tfrac{x}{2}\right)\right|\right)+C$
• $\int\sec(x)dx=\ln\big(|\sec(x)+\tan(x)|\big)+C$
• $\int\cot(x)dx=\ln\big(|\sin(x)|\big)+C$
• $\int\dfrac{dx}{\sqrt{1-x^2}}=\arcsin(x)+C$
• $\int-\dfrac{dx}{\sqrt{1-x^2}}=\arccos(x)+C$
• $\int\dfrac{dx}{1+x^2}=\arctan(x)+C$
• $\int -{\dfrac {dx}{1+x^{2}}}=\operatorname {arccot}(x)+C$
• $\int {\dfrac {dx}{x{\sqrt {x^{2}-1}}}}=\operatorname {arcsec} {\big (}|x|{\big )}+C$
• $\int -{\dfrac {dx}{x{\sqrt {x^{2}-1}}}}=\operatorname {arccsc} {\big (}|x|{\big )}+C$

Hyper Trigonometric functions

• $\int \sinh(x) dx = \cosh(x)+C$
• $\int \cosh(x) dx = \sinh(x)+C$
• $\int \tanh(x) dx = \ln(\cosh(x))+C$
• $\int \operatorname{csch}(x) dx = \ln\left(\left|\tanh\left(\tfrac{x}{2}\right)\right|\right)+C$
• $\int \operatorname{sech}(x) dx = 2\arctan\left(\tanh\left(\tfrac{x}{2}\right)\right)+C$
• $\int \operatorname{coth}(x) dx = \ln( \sinh (x))+C$
• $\int \dfrac{dx}{\sqrt{1+x^2}} = \operatorname{arcsinh}(x)+C$
• $\int \dfrac{dx}{\sqrt{x^2 -1}} = \operatorname{arccosh}(x)+C$
• $\int \dfrac{dx}{1-x^2} = \operatorname{arctanh}(x)+C$
• $\int \dfrac{dx}{1-x^2} = \operatorname{arccoth}(x)+C$
• $\int -\dfrac{dx}{x\sqrt{1-x^2}} = \operatorname{arcsech}\big(|x|\big)+C$
• $\int -\dfrac{dx}{x\sqrt{1+x^2}} = \operatorname{arccsch}\big(|x|\big)+C$