The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\sin ^{2} x}{1+2^{x}} \mathrm{~d} x$ is :

JEE Main · 2018 · 08 Apr 2018 · Offline · Definite Integrals

The value of

\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{s i n ^{2} x}{1 + 2 ^{x}} d x

is :

Answer: (A)

Apply the substitution

x \to - x

, which keeps the symmetric limits and turns the integrand into

\frac{s i n ^{2} x \cdot 2 ^{x}}{1 + 2 ^{x}}

. Adding this companion integral to the original, the two numerators combine to

sin^{2} x

, so

2 I = \int_{- π /2}^{π /2} sin^{2} x d x

. Since

sin^{2} x

is even,

2 I = 2 \int_{0}^{π /2} \frac{1 - c o s 2 x}{2} d x = [x - \frac{s i n 2 x}{2}]_{0}^{π /2} = \frac{π}{2}

, and therefore

I = \frac{π}{4}

JEE Main · 2018 · 08 Apr 2018 · Offline · Single correct option

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