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)}}dx $ is equal to :
(where c is a constant of integration)

• A. $\log_{e} \left|\sec\left(\frac{x^{2} -1}{2}\right)\right|+c $
• B. $\log_{e} \left| \frac{1}{2} \sec^{2} \left(x^{2}-1\right)\right|+c$
• C. $\frac{1}{2} \log_{e} \left|\sec^{2} \left(\frac{x^{2} -1}{2}\right)\right|+c $
• D. $\frac{1}{2} \log_{e} \left|\sec\left(x^{2} -1\right)\right|+c $

Answer 1

Answer: (A)

Put $\left(x^{2} -1\right) = 1 $
$ \Rightarrow 2xdx =dt $
$ \therefore I = \frac{1}{2} \int \sqrt{\frac{1-\cos t}{1+ \cos t}} dt $
$ = \frac{1}{2} \int\tan \left(\frac{t}{2}\right)dt $
$= ln \left|\sec \frac{t}{2}\right| +c $
$ I = ln \left|\sec \left(\frac{x^{2}-1}{2}\right)\right|+c $