JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $\alpha > 0$ . If $\int_{0}^{\alpha} \frac{x}{\sqrt{x + \alpha} - \sqrt{x}} dx = \frac{16 + 20\sqrt{2}}{15}$ , then $\alpha$ is equal to:

Choose the correct answer:

Explanation

Solution

Is integral ko solve karne ke liye pehle denominator ko rationalize karenge:

\frac{x}{\sqrt{x + \alpha} - \sqrt{x}} \times \frac{\sqrt{x + \alpha} + \sqrt{x}}{\sqrt{x + \alpha} + \sqrt{x}} = \frac{x(\sqrt{x + \alpha} + \sqrt{x})}{(x + \alpha) - x} = \frac{x(\sqrt{x + \alpha} + \sqrt{x})}{\alpha}

Ab integrate karte hain:

\frac{1}{\alpha} \int_{0}^{\alpha} (x\sqrt{x+\alpha} + x\sqrt{x}) dx

Is integral ko solve karke aur value $\frac{16 + 20\sqrt{2}}{15}$ ke barabar rakhne par humein $\alpha$ ki value milti hai. Calculation ke baad:

Sahi Answer: (4) 2

Choose the correct answer:

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