#### Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise  Question 88 Maths Textbook Solution.

$\left(\frac{\sqrt{3}}{2}\right)\left[\left(\frac{3 x-1}{3}\right) \sqrt{\frac{1}{3}+\frac{2 x}{3}-x^{2}}+\frac{4 \sin ^{-1}(3 x-1)}{9}\right]+c$

Hint:

You must know about formula of $\sqrt{a^{2}-x^{2}}, \sqrt{x^{2}+a^{2}}, \sqrt{x^{2}-a^{2}}$

Given:

$\int \sqrt{1+2 x-3 x^{2}} d x$

Solution:

$\int \sqrt{3} \sqrt{\frac{1}{3}+\frac{2}{3} x-x^{2}} d x$

$\sqrt{3} \int \frac{1}{3}+\frac{1}{9}-\frac{1}{9}+\frac{2 x}{3}-x^{2} d x$

$\int \sqrt{3} \sqrt{\frac{4}{9}-\frac{1}{9}-\frac{2}{3} x+x^{2}} d x$

$\int \sqrt{3} \sqrt{\left(\frac{2}{3}\right)^{2}-\left(x-\frac{1}{3}\right)^{2}} d x$                                    $\left[\because\left(\sqrt{a^{2}-x^{2}} d x=\frac{1}{2}\left(x \sqrt{a^{2}-x^{2}}\right)+a^{2} \sin ^{-1}\left(\frac{x}{a}\right)\right]\right.$

$\sqrt{3} \cdot\left(\frac{1}{2}\right)\left[\left(x-\frac{1}{3}\right) \sqrt{\left(\frac{2}{3}\right)^{2}-\left(x-\frac{1}{3}\right)^{2}}+\left(\frac{2}{3}\right)^{2} \frac{\sin ^{-1}\left(x-\frac{1}{8}\right)}{\frac{2}{3}}\right]$

$I=\left(\frac{\sqrt{3}}{2}\right)\left[\left(\frac{3 x-1}{3}\right) \sqrt{\frac{1}{3}+\frac{2 x}{3}-x^{2}}+\frac{4}{9} \frac{\sin ^{-1}(3 x-1)}{2}\right]+c$