$\displaystyle \int{\dfrac{1}{ax \pm b} \,}dx \,=\, \dfrac{1}{a}\log_e{|ax \pm b|}+c$

Let $a$ and $b$ represent constants, and $x$ represents a variable. The three liters form linear expression in one variable possibly in the following two ways.

- $ax+b$
- $ax-b$

Hence, the linear expression in one variable simply written in mathematical form as $ax \pm b$.

The indefinite integral for the reciprocal of the linear expression in one variable $ax\pm b$ with respect to $x$ is expressed mathematically as follows.

$\implies$ $\displaystyle \int{\dfrac{1}{ax \pm b} \,}dx$

The indefinite integral for the multiplicative inverse of the linear expression in one variable with respect to $x$ is equal to the product of the reciprocal of the coefficient of variable and the natural logarithm of the linear expression in one variable and the integral constant.

$(1) \,\,\,$ $\displaystyle \int{\dfrac{1}{ax+b} \,}dx \,=\, \dfrac{1}{a}\ln{|ax+b|}+c$

$(2) \,\,\,$ $\displaystyle \int{\dfrac{1}{ax-b} \,}dx \,=\, \dfrac{1}{a}\ln{|ax-b|}+c$

Evaluate $\displaystyle \int{\dfrac{1}{3x+2} \,}dx$

In this simple problem, $a = 3$ and $b = 2$.

$\therefore \,\,\,\,\,\,$ $\displaystyle \int{\dfrac{1}{3x+2} \,}dx \,=\, \dfrac{1}{3}\ln{|3x+2|}+c$

Learn how to derive the indefinite integral rule for the multiplicative inverse of the linear expression in one variable.

Latest Math Topics

Nov 11, 2022

Nov 03, 2022

Jul 24, 2022

Jul 15, 2022

Latest Math Problems

Nov 25, 2022

Nov 02, 2022

Oct 26, 2022

Oct 24, 2022

Sep 30, 2022

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 - 2022 Math Doubts, All Rights Reserved