The value of the integral $\int_{-\pi/2}^{\pi/2} \frac{dx}{(1 + e^{x})(\sin^{6}x + \cos^{6}x)}$ is equal to

JEE Main · 2022 · 24 Jun 2022 · Shift 2 · Definite Integrals

The value of the integral

\int_{- π /2}^{π /2} \frac{d x}{( 1 + e ^{x} ) ( s i n ^{6} x + c o s ^{6} x )}

is equal to

Answer: (C)

Using the property

\int_{- a}^{a} f (x) d x = \int_{0}^{a} [f (x) + f (- x)] d x

and substitution

x \to - x

I = \int_{0}^{π /2} \frac{d x}{s i n ^{6} x + c o s ^{6} x}

Note that

sin^{6} x + cos^{6} x = (sin^{2} x + cos^{2} x)^{3} - 3 sin^{2} x cos^{2} x (sin^{2} x + cos^{2} x) = 1 - 3 sin^{2} x cos^{2} x

Using

sin^{2} x cos^{2} x = \frac{s i n ^{2} ( 2 x )}{4}

, we get

sin^{6} x + cos^{6} x = 1 - \frac{3 s i n ^{2} ( 2 x )}{4}

After substitution

tan x = t

and simplification:

I = \int_{- \infty}^{\infty} \frac{d z}{1 + z ^{2}} = [tan^{- 1} z]_{- \infty}^{\infty} = \frac{π}{2} - (- \frac{π}{2}) = π

JEE Main · 2022 · 24 Jun 2022 · Shift 2 · Single correct option