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

The integral of reciprocal of the square root of difference of squares is equal to the sum of natural logarithm of sum of variable and square root of difference of squares, and the constant of integration.

Let $x$ be a variable and $a$ be a constant. The square root of the difference of their squares forms an irrational function in the following form.

$\sqrt{x^2-a^2}$

In some cases, this irrational function also appears in reciprocal form as follows.

$\dfrac{1}{\sqrt{x^2-a^2}}$

The indefinite integral of this type of irrational function in reciprocal form with respect to $x$ is written as follows.

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

The integral of one by square root of $x$ squared minus $a$ squared with respect to $x$ is called the integral rule for the square root of difference of squares in reciprocal form.

The integral of this irrational function is equal to the natural logarithm of $x$ plus square root of $x$ squared minus $a$ squared, and plus integral constant $c$.

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

According to the representation of natural logarithms, the natural logarithm can also be written in following form in mathematics.

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

Evaluate $\displaystyle \int{\dfrac{1}{\sqrt{x^2-16}}}\,dx$

In this example problem, the number $16$ can be expressed in square form.

$=\,\,\,$ $\displaystyle \int{\dfrac{1}{\sqrt{x^2-4^2}}}\,dx$

Take $a \,=\, 4$ and substitute it in the integral rule for the square root of difference of squares in reciprocal form.

$=\,\,\,$ $\log_{e}{\Big|x+\sqrt{x^2-4^2}\Big|}$ $+$ $c$

$=\,\,\,$ $\log_{e}{\Big|x+\sqrt{x^2-16}\Big|}$ $+$ $c$

Learn how to derive the integral rule for the square root of difference of squares in reciprocal form.

Latest Math Topics

Jan 06, 2023

Jan 03, 2023

Jan 01, 2023

Dec 26, 2022

Dec 08, 2022

Latest Math Problems

Jan 31, 2023

Nov 25, 2022

Nov 02, 2022

Oct 26, 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