Integration Rules

Properties

$\displaystyle \int udv$ $\,=\,$ $uv$ $-$ $\displaystyle \int vdu$

Formulas

There are some standard results formed by some functions in integral calculus

Integration of Algebraic functions

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

$(2)\,\,\,$ $\displaystyle \int{\dfrac{1}{x}\,}dx$ $\,=\,$ $\log_{e}{x}+c$

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

$(4)\,\,\,$ $\displaystyle \int{e^x\,}dx$ $\,=\,$ $e^x+c$

$(5)\,\,\,$ $\displaystyle \int{\dfrac{1}{x^2-a^2}\,}dx$ $\,=\,$ $\dfrac{1}{2a}\log_{e}{\Bigg|\dfrac{x-a}{x+a}\Bigg|}+c$

$(6)\,\,\,$ $\displaystyle \int{\dfrac{1}{x^2+a^2}\,}dx$ $\,=\,$ $\dfrac{1}{a}\tan^{-1}{\Big(\dfrac{x}{a}\Big)}+c$

Integration of Trigonometric functions

$\Large \int \normalsize \sin{x} dx = -\cos{x}+c$

$\Large \int \normalsize \cos{x} dx = \sin{x}+c$

$\Large \int \normalsize \tan{x} dx = -\log{(\cos{x})}+c$

$\Large \int \normalsize \cot{x} dx = \log{(\sin{x})}+c$

$\Large \int \normalsize \sec^2{x} dx = \tan{x}+c$

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

$\Large \int \normalsize \sec{x}\tan{x} dx = \sec{x}+c$

$\Large \int \normalsize \csc{x}\cot{x} dx = -\csc{x}+c$

Integration of Hyperbolic functions

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

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

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

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

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

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

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

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

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

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

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