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Integrate the function
$\int \frac{2+\sin 2 x}{1+\cos 2 x} e^x d x=I(s a y)$
$\int \frac{2+\sin 2 x}{1+\cos 2 x} e^x d x=I(s a y)$
Solution:
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Verified Answer
$\int \frac{2+2 \sin x \cos x}{2 \cos ^2 x} e^x d x=\int e^x\left(\sec ^2 x+\tan x\right) d x$
Put $e^x \tan x=t$ so that $\left(e^x \sec ^2 x+\tan x e^x\right) d x=d t$
$\Rightarrow \quad e^x\left(\sec ^2 x+\tan x\right) d x=d t$
$\therefore \quad I=\int d t=t+c=e^x \tan x+c$
Put $e^x \tan x=t$ so that $\left(e^x \sec ^2 x+\tan x e^x\right) d x=d t$
$\Rightarrow \quad e^x\left(\sec ^2 x+\tan x\right) d x=d t$
$\therefore \quad I=\int d t=t+c=e^x \tan x+c$
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