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Let \(I(x)=\int \frac{x^2\left(x \sec ^2 x+\tan x\right)}{(x \tan x+1)^2} d x\) If \(I(0)=0\) then \(I\left(\frac{\pi}{4…

Q1

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

[JEE Main 2023, 6 Apr (Shift 1)]

a

\(\log _e \frac{(\pi+4)^2}{16}-\frac{\pi^2}{4(\pi+4)}\)

b

\(\log _e \frac{(\pi+4)^2}{16}+\frac{\pi^2}{4(\pi+4)}\)

c

\(\log _e \frac{(\pi+4)^2}{32}-\frac{\pi^2}{4(\pi+4)}\)

d

\(\log _e \frac{(\pi+4)^2}{32}+\frac{\pi^2}{4(\pi+4)}\)

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