Let I(x) = ∫ x²(xsec² x + tan x)/(xtan x + 1)² dx. If I(0) = 0, t… | JEE Main 2023 Mathematics PYQ | Jeeify

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MCQMediumJEE Main 2023 · 6 April, Shift 1Integration Techniques (Substitution, Parts, Partial Fractions)

Let $I(x) = \int \frac{x^2(x\sec^2 x + \tan x)}{(x\tan x + 1)^2} \, dx$ . If $I(0) = 0$ , then $I\left(\frac{\pi}{4}\right)$ is equal to:

A
$\log_e\left(\frac{(\pi+4)^2}{16}\right) + \frac{\pi^2}{4(\pi+4)}$
B
$\log_e\left(\frac{(\pi+4)^2}{32}\right) - \frac{\pi^2}{4(\pi+4)}$
C
$\log_e\left(\frac{(\pi+4)^2}{16}\right) - \frac{\pi^2}{4(\pi+4)}$
D
$\log_e\left(\frac{(\pi+4)^2}{32}\right) + \frac{\pi^2}{4(\pi+4)}$

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