For $x^{2} \neq \mathrm{n} \pi+1, \mathrm{n} \in \mathrm{N}$ (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 $c$ is a constant of integration)
Correct Option: , 3
Consider the given integral
$I=\int x \sqrt{\frac{2 \sin \left(x^{2}-1\right)-2 \sin \left(x^{2}-1\right) \cos \left(x^{2}-1\right)}{2 \sin \left(x^{2}-1\right)+2 \sin \left(x^{2}-1\right) \cos \left(x^{2}-1\right)}} d x$
$(\therefore \sin 2 \theta=2 \sin \theta \cos \theta)$
$\Rightarrow \quad I=\int x \sqrt{\frac{1-\cos \left(x^{2}-1\right)}{1+\cos \left(x^{2}-1\right)}} d x$
$\Rightarrow I=\int x\left|\tan \left(\frac{x^{2}-1}{2}\right)\right| d x$
Now let $\frac{x^{2}-1}{2}=t \quad \Rightarrow \quad \frac{2 x}{2} d x=d t$
$\therefore \quad I=\int|\tan (t)| d t=\ln |\sec t|+C$
or $\quad I=\ln \left|\sec \left(\frac{x^{2}-1}{2}\right)\right|+c=\frac{1}{2} \ln \left|\sec ^{2}\left(\frac{x^{2}-1}{2}\right)\right|+c$
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