If the value of the integral - 1 1 cos x 1 + 3 x ? d x is 2. Then, a value of is [2024]

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Solution

Q.
If the value of the integral $\int_{- 1}^{1} \frac{\cos α x}{1 + 3^{x}} d x$ is $\frac{2}{π} .$ Then, a value of $α$ is [2024]

1 $\frac{π}{6}$

2 $\frac{π}{3}$

3 $\frac{π}{2}$

4 $\frac{π}{4}$

Ans.
(3)

Let $I = \int_{- 1}^{1} \frac{\cos α x}{1 + 3^{x}} d x$ ....(i)

$= \int_{- 1}^{1} \frac{\cos α (1 - 1 - x)}{1 + 3^{1 - 1 - x}} d x$

$(using property \int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x)$

$\Rightarrow I = \int_{- 1}^{1} \frac{\cos α x}{1 + 3^{- x}} d x = \int_{- 1}^{1} \frac{3^{x} \cos α x}{1 + 3^{x}} d x$ ...(ii)

On adding (i) and (ii), we get

$2 I = \int_{- 1}^{1} \cos (α x) d x = {(\frac{\sin (α x)}{α})}_{- 1}^{1}$

$\Rightarrow 2 (\frac{2}{π}) = \frac{2 \sin α}{α} [∵ I = \int_{- 1}^{1} \frac{\cos α x}{1 + 3^{x}} d x = \frac{2}{π} (Given)]$

$\Rightarrow \frac{\sin α}{α} = \frac{2}{π} \Rightarrow α = \frac{π}{2} .$

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