Math Doubts

Integral rule for Reciprocal of difference of squares


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


Let $x$ represents a variable and $a$ represents a constant. The subtraction of square of $a$ from $x$ squared is written as $x^2-a^2$ in mathematics. The multiplicative inverse of the difference of them is written in the following mathematical form.


The indefinite integral of this rational expression with respect to $x$ is written in calculus as follows.

$\implies$ $\displaystyle \int{\dfrac{1}{x^2-a^2}\,}dx$

The indefinite integration of this rational function is equal to product of the reciprocal of the two times the constant and the natural logarithm of the quotient of the difference by sum of them.

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

Alternative forms

The indefinite integral formula for multiplicative inverse of the difference of squares can be written in terms of any two literals.

$(1) \,\,\,$ $\displaystyle \int{\dfrac{1}{y^2-m^2}\,}dy \,=\, \dfrac{1}{2m}\log_e{\Bigg|\dfrac{y-m}{y+m}\Bigg|}+c$

$(2) \,\,\,$ $\displaystyle \int{\dfrac{1}{l^2-d^2}\,}dl \,=\, \dfrac{1}{2d}\log_e{\Bigg|\dfrac{l-d}{l+d}\Bigg|}+c$

$(3) \,\,\,$ $\displaystyle \int{\dfrac{1}{z^2-f^2}\,}dz \,=\, \dfrac{1}{2f}\log_e{\Bigg|\dfrac{z-f}{z+f}\Bigg|}+c$


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

Take $a = 3$ and substitute it in the integral property.

$=\,\,\,$ $\dfrac{1}{2 \times 3}\log_e{\Bigg|\dfrac{x-3}{x+3}\Bigg|}+c$

$=\,\,\,$ $\dfrac{1}{6}\log_e{\Bigg|\dfrac{x-3}{x+3}\Bigg|}+c$


Learn how to derive the indefinite integration rule for multiplicative inverse of difference of squares.

Math Doubts
Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more