If the value of the integral $\int_{-1}^{1}\dfrac{\cos\alpha x}{1+3^{x}}dx$ is $\dfrac{2}{\pi}$. Then, a value of $\alpha$ is

JEE Main · 2024 · 04 Apr 2024 · Shift 2 · Definite Integrals

If the value of the integral

\int_{- 1}^{1} \frac{cos α x}{1 + 3 ^{x}} d x

\frac{2}{π}

. Then, a value of

α

Answer: (D)

I = \int_{- 1}^{1} \frac{cos α x}{1 + 3 ^{x}} d x

. Using

x \to - x

I = \int_{- 1}^{1} \frac{cos α x \cdot 3 ^{x}}{1 + 3 ^{x}} d x

(since

cos

is even).

Adding the two forms of

I

2 I = \int_{- 1}^{1} cos α x d x = \frac{2 sin α}{α} \Rightarrow I = \frac{sin α}{α}

.

Given

I = \frac{2}{π}

: testing

α = \frac{π}{2}

gives

\frac{sin ( π /2 )}{π /2} = \frac{1}{π /2} = \frac{2}{π}

✓.

JEE Main · 2024 · 04 Apr 2024 · Shift 2 · Single correct option

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