Question

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

• A. 2
• B. 1
• C. 4
• D. 0

Answer 1

Answer: (B)

Let $I =\int\limits_{-\pi / 4}^{\pi / 4} \frac{d x}{1+\cos \,2 x} $
$=\int\limits_{-\pi / 4}^{\pi / 4} \frac{d x}{2 \cos ^{2} x} $
$=\frac{1}{2} \int\limits_{-\pi / 4}^{\pi / 4} \sec ^{2} x\, d x$
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} \int\limits_{0}^{\pi / 4} \sec ^{2} x d x=[\tan x]_{0}^{\pi / 4} $
$=\tan \frac{\pi}{4}-\tan 0=1-0=1$