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

Jun 05, 2023

May 21, 2023

May 16, 2023

May 10, 2023

May 03, 2023

Latest Math Problems

May 09, 2023

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

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

A math help place with list of solved problems with answers and worksheets on every concept for your practice.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved