The value of $\int_{-\pi}^{\pi} \frac{\cos^{2} x}{1+a^{x}} dx, a>0$, is

JEE Main · 2005 · 08 May 2005 · Offline · Definite Integrals

The value of

\int_{- π}^{π} \frac{c o s ^{2} x}{1 + a ^{x}} d x, a > 0

, is

Answer: (B)

Let

I = \int_{- π}^{π} \frac{cos ^{2} x}{1 + a ^{x}} d x

. Replacing

x

- x

gives

I = \int_{- π}^{π} \frac{cos ^{2} x}{1 + a ^{- x}} d x = \int_{- π}^{π} \frac{a ^{x} cos ^{2} x}{1 + a ^{x}} d x

. Adding the two forms, the denominators combine to give

2 I = \int_{- π}^{π} cos^{2} x d x

. Since

\int_{- π}^{π} cos^{2} x d x = π

, we obtain

I = \frac{π}{2}

JEE Main · 2005 · 08 May 2005 · Offline · Single correct option

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