JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The value of $\int_{\pi/3}^{\pi/2} \frac{(2 + 3\sin x)}{\sin x (1 + \cos x)} dx$ is equal to:

Choose the correct answer:

Explanation

Solution

Split the integral:

I = \int_{\pi/3}^{\pi/2} \frac{2}{\sin x (1 + \cos x)} dx + \int_{\pi/3}^{\pi/2} \frac{3}{1 + \cos x} dx

For the second part: $\frac{3}{1 + \cos x} = \frac{3}{2 \cos^2(x/2)} = \frac{3}{2} \sec^2(x/2)$ .

$\int \frac{3}{2} \sec^2(x/2) dx = 3 \tan(x/2)$ .

For the first part, use substitution $t = \tan(x/2)$ , where $\sin x = \frac{2t}{1+t^2}$ and $\cos x = \frac{1-t^2}{1+t^2}$ .

After partial fractions and integration, and applying limits $[\pi/3, \pi/2]$ (where $t$ goes from $1/\sqrt{3}$ to $1$ ):

Final result calculation leads to:

I = \frac{10}{3} - \sqrt{3} - \log_e \sqrt{3}

Correct Option: (1)

Choose the correct answer:

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