∫ex2+sin2x1+cos2xdx=

∫ex2+sin2x1+cos2xdx=∫ex(2+2sinxcosx)2cos2xdx=∫ex1+sinxcosxcos2xdx=∫ex(sec2x+tanx)f(x)=Tanx,f1(x)=sec2x=exTanx+c

∫ex2+sin2x1+cos2xdx=

∫ex2+sin2x1+cos2xdx=∫ex(2+2sinxcosx)2cos2xdx=∫ex1+sinxcosxcos2xdx=∫ex(sec2x+tanx)f(x)=Tanx,f1(x)=sec2x=exTanx+c

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$\int e^{x} (\frac{2 + \sin 2 x}{1 + \cos 2 x}) d x =$

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$2 e^{x} \sec^{2} x + c$

$e^{x} \cos 2 x + c$

$e^{x} \cot x + c$

$e^{x} \tan x + c$

answer is D.

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Detailed Solution

$\int e^{x} (\frac{2 + \sin 2 x}{1 + \cos 2 x}) d x = \int e^{x} \frac{(2 + 2 \sin x \cos x)}{2 \cos^{2} x} d x$

$= \int e^{x} (\frac{1 + \sin x \cos x}{\cos^{2} x}) d x = \int e^{x} (\sec^{2} x + \tan x)$

$[f (x) = T a n x, f^{1} (x) = \sec^{2} x]$

$= e^{x} T a n x + c$

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$\int e^{x} (\frac{2 + \sin 2 x}{1 + \cos 2 x}) d x =$