Question

$\underset{-\pi /4}{\overset{\pi /4}{\int }}$ $\frac{dx}{1+\cos 2x}$ is equal to

• A. 2
• B. 1
• C. 4
• D. 0

Answer 1

Answer: (B)

Let $I=\underset{-\pi /4}{\overset{\pi /4}{\int }}\frac{dx}{1+\cos 2x}$
$=\underset{-\pi /4}{\overset{\pi /4}{\int }}\frac{dx}{2{\cos }^{2}x}$
$=\frac{1}{2}\underset{-\pi /4}{\overset{\pi /4}{\int }}{\sec }^{2}xdx$
Let $f(x)={\sec }^{2}x$
$\Rightarrow f(-x)={\sec }^2(-x)$
$={\sec }^{2}x=f(x)$
Hence, $f(x)$ is an even function.
$\therefore I=\frac{2}{2}\underset{0}{\overset{\pi /4}{\int }}{\sec }^{2}xdx=[\tan x{]}_0^{\pi /4}$
$=\tan \frac{\pi }{4}-\tan 0=1-0=1$