Math Doubts

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

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

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

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

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

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

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

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

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

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

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

Math Doubts

A best free mathematics education website for students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved