JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The value of the integral $\int_{-\pi/4}^{\pi/4} \frac{x + \pi/4}{2 - \cos 2x} dx$ is:

Choose the correct answer:

Explanation

Solution

Integral ko do bhagon mein baantein: $I = \int_{-\pi/4}^{\pi/4} \frac{x}{2 - \cos 2x} dx + \int_{-\pi/4}^{\pi/4} \frac{\pi/4}{2 - \cos 2x} dx$ .

Pehla bhag odd function hai, isliye uska value $0$ ho jayega.

Doosra bhag even function hai:

$I = 2 \times \frac{\pi}{4} \int_{0}^{\pi/4} \frac{1}{2 - \cos 2x} dx = \frac{\pi}{2} \int_{0}^{\pi/4} \frac{\sec^2 x}{1 + 3\tan^2 x} dx$ .

Substitute karein $\sqrt{3}\tan x = t$ :

$I = \frac{\pi}{2\sqrt{3}} \int_{0}^{\sqrt{3}} \frac{1}{1 + t^2} dt = \frac{\pi}{2\sqrt{3}} [\tan^{-1} t]_0^{\sqrt{3}} = \frac{\pi}{2\sqrt{3}} \times \frac{\pi}{3} = \frac{\pi^2}{6\sqrt{3}}$ .

Correct Option: (2)

Choose the correct answer:

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