Integrate the function 2+ 2 x 1+ 2 x e^ x - Vidyadip

Question

Integrate the function $\frac{2+\sin 2 x}{1+\cos 2 x} e^{x}$

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Answer

Let $I=\frac{2+\sin 2 x}{1+\cos 2 x} e^{x}$
$= \left(\frac{2+2 \sin x \cos x}{2 \cos ^{2} x}\right) e^{x}$
$= 2 \cdot\left(\frac{1+\sin x \cos x}{2 \cos ^{2} x}\right) e^{x}$
$= \left(\frac{1}{\cos ^{2} x}+\frac{\sin x \cos x}{\cos ^{2} x}\right) e^{x}$
$= \left(\sec ^{2} x+\tan x\right) e^{x}$
$\Rightarrow \int \frac{2+\sin 2 x}{1+\cos 2 x} e^{x} d x=\int\left(\sec ^{2} x+\tan x\right) e^{x} d x$
Now let $\tan x = f(x)$
$\Rightarrow f'(x) = \sec^2x\ dx $
$\Rightarrow \int\left(\sec ^{2} x+\tan x\right) e^{x} d x = \int\left(f(x)+f^{\prime}(x)\right) e^{x} d x = e^{x} f(x)+C$
$\Rightarrow I=e^{x} \tan x+C$

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