Question

$\underset{0}{\overset{{\pi }^2/4}{\int }}\left(\frac{2}{\csc \sqrt{x}}+\frac{\sqrt{x}}{\sec \sqrt{x}}\right)dx$ is equal to

• A. $\frac{{\pi }^2}{4}$
• B. $\frac{{\pi }^2}{2}$
• C. ${\pi }^2$
• D. $\frac{\pi }{8}$

Answer 1

Answer: (B)

$I=\underset{0}{\overset{{\pi }^2/4}{\int }}(2\sin \sqrt{x}+\sqrt{x}\cos x)dx;$
$\mathrm{Let}f(x)=2\sin \sqrt{x};f^{\prime }(x)=\frac{\cos \sqrt{x}}{\sqrt{x}}$
$x{f}^{\prime }(x)=\sqrt{x}\cos \sqrt{x}$
$\therefore I=2x\sin \sqrt{x}{]}_0^{{\pi }^2/4}=2\cdot \frac{{\pi }^2}{4}=\frac{{\pi }^2}{2}$