Integration formulas

There are some fundamental integral rules and they are used as formulas in integral calculus. So, learn each formula with proof for studying the integral calculus clearly.

Algebraic functions

The list of integration formulas for the algebraic functions with proofs.

$(1)\,\,\,$ $\displaystyle \int{x^n\,}dx \,=\, \dfrac{x^{n+1}}{n+1}+c$

$(2)\,\,\,$ $\displaystyle \int{a^x\,}dx \,=\, \dfrac{a^x}{\log_e{|a|}}+c$

$(3)\,\,\,$ $\displaystyle \int{e^x\,}dx \,=\, e^x+c$

$(4)\,\,\,$ $\displaystyle \int{\dfrac{1}{x}\,}dx \,=\, \log_e{|x|}+c$

$(5)\,\,\,$ $\displaystyle \int{\dfrac{1}{ax\pm b}\,}dx \,=\, \dfrac{1}{a}\log_e{|ax\pm b|}+c$

Trigonometric functions

The list of integration formulas for the trigonometric functions with proofs.

$(1)\,\,\,$ $\displaystyle \int{\sin{x}\,}dx \,=\, -\cos{x}+c$

$(2)\,\,\,$ $\displaystyle \int{\cos{x}\,}dx \,=\, \sin{x}+c$

$(3)\,\,\,$ $\displaystyle \int{\tan{x}\,}dx \,=\, -\log_e{|\cos{x}|}+c$

$(4)\,\,\,$ $\displaystyle \int{\cot{x}\,}dx \,=\, \log_e{|\sin{x}|}+c$

$(5)\,\,\,$ $\displaystyle \int{\sec^2{x}\,}dx \,=\, \tan{x}+c$

$(6)\,\,\,$ $\displaystyle \int{\csc^2{x}\,}dx \,=\, -\cot{x}+c$

$(7)\,\,\,$ $\displaystyle \int{\sec{x}\tan{x}\,}dx \,=\, \sec{x}+c$

$(8)\,\,\,$ $\displaystyle \int{\csc{x}\cot{x}\,}dx \,=\, -\csc{x}+c$

Hyperbolic functions

The list of integration formulas for the hyperbolic functions with proofs.

$\Large \int \normalsize \sinh{x} dx = \cosh{x}+C$

$\Large \int \normalsize \cosh{x} dx = \sinh{x}+C$

$\Large \int \normalsize \tanh{x} dx = \log_{e}{|\cosh{x}|}+C$

$\Large \int \normalsize \coth{x} dx = \log_{e}{|\sinh{x}|}+C$

$\Large \int \normalsize \operatorname{sech}{x} dx = 2\tan^{-1}{(e^x)}+C$

$\Large \int \normalsize \operatorname{csch}{x} dx = 2\cosh^{-1}{(e^x)}+C$

$\Large \int \normalsize \sec^2h{x} dx = \tanh{x}+C$

$\Large \int \normalsize \csc^2h{x} dx = -\cot{x}+C$

$\Large \int \normalsize \operatorname{sech}{x}\tanh{x} dx = -\operatorname{sech}{x}+C$

$\Large \int \normalsize \operatorname{csch}{x}\coth{x} dx = -\csc{x}+C$