Question

For $x^2\ne n\pi +1,n\in N$ (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 $\left(x^2-1\right)=1$
$\Rightarrow 2xdx=dt$
$\therefore I=\frac{1}{2}\int \sqrt{\frac{1-\cos t}{1+\cos t}}dt$
$=\frac{1}{2}\int \tan \left(\frac{t}{2}\right)dt$
$=ln\left|\sec \frac{t}{2}\right|+c$
$I=ln\left|\sec \left(\frac{x^2-1}{2}\right)\right|+c$