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JEE 2023 Mathematics PYQ — Let and . If , then is equal to ___.… | Mathem Solvex | Mathem Solvex

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $I(x) = \int \sqrt{\frac{x+7}{x}} dx$ and $I(9) = 12 + 7 \log_e 7$ . If $I(1) = \alpha + 7 \log_e (1+2\sqrt{2})$ , then $\alpha^4$ is equal to ___.

Choose the correct answer:

A.
64
(Correct Answer)
B.
65
C.
66
D.
67

Correct Answer:

Explanation

Solution:

Integral ko solve karne ke liye $x = t^2$ rakhein:

I(x) = \int \sqrt{x+7} \frac{dx}{\sqrt{x}} = \sqrt{x(x+7)} + 7 \log_e (\sqrt{x} + \sqrt{x+7}) + C

$I(9)$ se $C$ nikalna:

$I(9) = \sqrt{9(16)} + 7 \log_e (3+4) + C = 12 + 7 \log_e 7 + C$

Humein $12 + 7 \log_e 7$ diya gaya hai, isliye $C = 0$ .
$I(1)$ nikalna:

$I(1) = \sqrt{1(8)} + 7 \log_e (1+\sqrt{8}) = \sqrt{8} + 7 \log_e (1+2\sqrt{2})$

Iska matlab $\alpha = \sqrt{8}$ .
Final Answer:

$\alpha^4 = (\sqrt{8})^4 = 8^2 = 64$

Explanation

Solution:

Integral ko solve karne ke liye $x = t^2$ rakhein:

I(x) = \int \sqrt{x+7} \frac{dx}{\sqrt{x}} = \sqrt{x(x+7)} + 7 \log_e (\sqrt{x} + \sqrt{x+7}) + C

$I(9)$ se $C$ nikalna:

$I(9) = \sqrt{9(16)} + 7 \log_e (3+4) + C = 12 + 7 \log_e 7 + C$

Humein $12 + 7 \log_e 7$ diya gaya hai, isliye $C = 0$ .
$I(1)$ nikalna:

$I(1) = \sqrt{1(8)} + 7 \log_e (1+\sqrt{8}) = \sqrt{8} + 7 \log_e (1+2\sqrt{2})$

Iska matlab $\alpha = \sqrt{8}$ .
Final Answer:

$\alpha^4 = (\sqrt{8})^4 = 8^2 = 64$

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