In integral calculus, there are some notable integral rules in which the mathematical functions are in reciprocal form (multiplicatively inverse) and they are used as formulas in integration. So, let’s learn the list of reciprocal integration rules with proofs before using them in definite and indefinite integration problems.

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

The integral of the reciprocal of a variable is equal to the natural logarithm of variable plus the integral constant.

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

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

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

$\displaystyle \int{\dfrac{1}{\sqrt{1-x^2}}\,}dx$ $\,=\,$ $\arcsin{x}+c$ (or) $\sin^{-1}{x}+c$

The integral of one by square root of one minus square of a variable is equal to the inverse sine of the variable plus the integral constant.

$\displaystyle \int{\dfrac{1}{|x|\sqrt{x^2-1}}\,}dx$ $\,=\,$ $\operatorname{arcsec}{x}+c$ (or) $\sec^{-1}{x}+c$

Latest Math Topics

Latest Math Problems

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

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

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2021 Math Doubts, All Rights Reserved