Evaluate : e^x (2+ 2 x 1+ 2 x ) d x - Vidyadip

Question

Evaluate : $\int e^x\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right) \cdot d x$

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Answer

$
\begin{aligned}
\text { I } & =\int e^x\left(\frac{2+2 \sin x \cdot \cos x}{2 \cdot \cos ^2 x}\right) \cdot d x \\
& =\int e^x\left(\frac{1}{\cos ^2 x}+\frac{\sin x \cdot \cos x}{\cos ^2 x}\right) \cdot d x \\
& =\int e^x\left[\sec ^2 x+\tan x\right] \cdot d x \\
& =\int e^x\left[\tan x+\sec ^2 x\right] \cdot d x \\
& \therefore f(x)=\tan x \Rightarrow f^{\prime}(x)=\sec ^2 x \\
& \therefore \int e^x\left[f(x)+f^{\prime}(x)\right] \cdot d x=e^x \cdot f(x)+c \\
& =e^{x \cdot \tan x+c} \\
\therefore & \int e^x\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right) \cdot d x=e^x \cdot \tan x+c
\end{aligned}
$

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