Standard functions

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

Logarithmic and exponential functions

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

Trigonometric functions

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

See also

• Integration
• Table of derivatives