#### Please Solve R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.18 Question 10 Maths Textbook Solution.

Answer: $\log \left|\sin ^{2} x+2+\sqrt{\sin ^{4} x+4 \sin ^{2} x-2}\right|+c$

Hint Let $\sin ^{2} x=t$

Given: $\int \frac{\sin 2 x}{\sqrt{\sin ^{4} x+4 \sin ^{2} x-2}} d x$

Explanation:

$\int \frac{\sin 2 x}{\sqrt{\sin ^{4} x+4 \sin ^{2} x-2}} d x$.................(1)

Let  $\sin ^{2} x=t$

$2 \sin x \cos x d x=d t$

$\sin 2 x d x=d t$                                                  (Differentiate w.r.t to t)

From (1) we have

$\int \frac{d t}{\sqrt{t^{2}+4 t-2}}$

$=\int \frac{d t}{\sqrt{t^{2}+4 t+4-4-2}}$

$=\int \frac{d t}{\sqrt{(t+2)^{2}-6}}$

Let t+2 = u                                                                        (Differentiate w.r.t to u)

$\mathrm{dt}=\mathrm{du}$

$=\int \frac{d u}{\sqrt{u^{2}-6}}$

$=\log \left|u+\sqrt{u^{2}-6}\right|+c$

$=\log \left|t+2+\sqrt{(t+2)^{2}-6}\right|+c$

$=\log \left|\sin ^{2} x+2+\sqrt{\sin ^{4} x+4 \sin ^{2} x-2}\right|+c\left[\because \int \frac{d x}{\sqrt{x^{2}-a^{2}}}=\log \left|x+\sqrt{x^{2}-a^{2}}\right|+c\right]$