The value of $\int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x}{1+3^{x}} d x$ is :

JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Definite Integrals

The value of

\int_{- π /2}^{π /2} \frac{c o s ^{2} x}{1 + 3 ^{x}} d x

is :

Answer: (B)

Use the property

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

. Here,

f (x) = \frac{c o s ^{2} x}{1 + 3 ^{x}}

and

f (- x) = \frac{c o s ^{2} x}{1 + 3 ^{- x}}

. Adding:

f (x) + f (- x) = \frac{c o s ^{2} x}{1 + 3 ^{x}} + \frac{c o s ^{2} x}{1 + 3 ^{- x}} = \frac{c o s ^{2} x ( 1 + 3 ^{- x} + 1 + 3 ^{x} )}{( 1 + 3 ^{x} ) ( 1 + 3 ^{- x} )} = cos^{2} x

. Thus,

I = \int_{0}^{π /2} cos^{2} x d x = \int_{0}^{π /2} \frac{1 + c o s 2 x}{2} d x = \frac{1}{2} [x + \frac{s i n 2 x}{2}]_{0}^{π /2} = \frac{π}{4}

JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Single correct option

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