# 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.

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

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

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

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

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

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

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

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

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

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

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