# Integral rule for Reciprocal of Linear expression

## Formula

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

### Introduction

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.

1. $ax+b$
2. $ax-b$

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.

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

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

#### Example

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

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

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

#### Proof

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

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Jun 26, 2023

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