Question

$\int\limits_0^{\pi^2 / 4}\left(\frac{2}{\csc \sqrt{x}}+\frac{\sqrt{x}}{\sec \sqrt{x}}\right) d x$ is equal to

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

Answer 1

Answer: (B)

$I=\int\limits_0^{\pi^2 / 4}(2 \sin \sqrt{x}+\sqrt{x} \cos x) d x ;$
$\operatorname{Let} f(x)=2 \sin \sqrt{x} ; f^{\prime}(x)=\frac{\cos \sqrt{x}}{\sqrt{x}}$
$x f^{\prime}(x)=\sqrt{x} \cos \sqrt{x} $
$\therefore I=2 x \sin \sqrt{x}]_0^{\pi^2 / 4}=2 \cdot \frac{\pi^2}{4}=\frac{\pi^2}{2}$