#### Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise  Question 86 Maths Textbook Solution.

$a^{2}\left(\frac{1}{2} \sin ^{-1}\left(\frac{x}{a}\right)+\left(\frac{x}{a}\right) \cdot \sqrt{a^{2}-x^{2}}\right)+c$

Hint:

You must know about how to solve integration.

Given:

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

Solution:

$\text { let } x=a \sin \theta$

$d x=a \cos \theta d \theta$

$I=\int \sqrt{a^{2}-a^{2} \sin ^{2} \theta} d \theta$

$I=a^{2} \int \sqrt{1-\sin ^{2} \theta} \cdot \cos \theta d \theta \quad \sqrt{1-\sin ^{2} \theta}=\cos ^{2} \theta$

$I=a^{2}\left(\frac{\theta}{2}+\frac{1}{2} \frac{(\sin 2 \theta)}{2}\right)+c$                                        $\left[\operatorname{let} \theta=\sin ^{-1}\left(\frac{x}{a}\right), \cos \theta=\sqrt{1-\left(\frac{x^{2}}{a^{2}}\right)}\right.$

$I=a^{2}\left(\frac{1}{2} \sin ^{-1}\left(\frac{x}{a}\right)+\frac{1}{2} \frac{2(\sin 2 \theta) \cos \theta}{2}\right)+c$

$I=a^{2}\left(\frac{1}{2} \sin ^{-1}\left(\frac{x}{a}\right)+\frac{1}{2}\left(\frac{x}{a}\right) \cdot \frac{1}{a} \sqrt{a^{2}-x^{2}}\right)+c$

$I=a^{2}\left(\frac{1}{2} \sin ^{-1}\left(\frac{x}{a}\right)+\left(\frac{x}{a}\right) \cdot \sqrt{a^{2}-x^{2}}\right)+c$