Question:
For $x^{2} \neq n \pi+1, n \in N$ (the set of natural numbers), the integral
$\int x \sqrt{\frac{2 \sin \left(x^{2}-1\right)-\sin 2\left(x^{2}-1\right)}{2 \sin \left(x^{2}-1\right)+\sin 2\left(x^{2}-1\right)}} d x$
is equal to :
(where $\mathrm{c}$ is a constant of integration)
Correct Option: 1
Solution:
Put $\left(x^{2}-1\right)=1$
$\Rightarrow 2 x d x=d t$
$\therefore \quad \mathrm{I}=\frac{1}{2} \int \sqrt{\frac{1-\cos t}{1+\cos t}} d t$
$=\frac{1}{2} \int \tan \left(\frac{\mathrm{t}}{2}\right) \mathrm{dt}$
$=\ln \left|\sec \frac{\mathrm{t}}{2}\right|+\mathrm{c}$
$I=\ln \left|\sec \left(\frac{x^{2}-1}{2}\right)\right|+c$
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