∫ex2+sin⁡2x1+cos⁡2xdx=

=∫ex2+sin⁡2x1+cos⁡2xdx=∫ex2+sin⁡2x1+2cos2⁡x−1dx=∫ex2(1+cos⁡xsin⁡x)2cos2⁡xdx=∫extan⁡x+sec2⁡xdx=extan⁡x+C

∫ex2+sin⁡2x1+cos⁡2xdx=

=∫ex2+sin⁡2x1+cos⁡2xdx=∫ex2+sin⁡2x1+2cos2⁡x−1dx=∫ex2(1+cos⁡xsin⁡x)2cos2⁡xdx=∫extan⁡x+sec2⁡xdx=extan⁡x+C

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

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$e^{x} cosec x + C$

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

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

$e^{x} \sec x + C$

answer is B.

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Detailed Solution

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

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

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

$= \int e^{x} [\tan x + \sec^{2} x] d x$

$= e^{x} \tan x + C$

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∫ex2+sin⁡2x1+cos⁡2xdx=

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