Question

For $x^2\ne n\pi +1,nϵN$ (the set of natural numbers), the integral
$\int x\sqrt{\frac{2\sin (x^2-1)-\sin 2(x^2-1)}{2\sin (x^2-1)+\sin 2(x^2-1)}}dx$ 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. ${\log }_e\left|\frac{1}{2}{\sec }^2\right(x^2-1\left)\right|+c$
• C. $\frac{1}{2}{\log }_e\left|{\sec }^2\left(\frac{x^2-1}{2}\right)\right|+c$
• D. $\frac{1}{2}{\log }_e\left|\sec \right(x^2-1\left)\right|+c$

Answer 1

The correct option is A ${\log }_e\left|\sec \left(\frac{x^2-1}{2}\right)\right|+c$

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