Evaluate the following functions : e^x(1+x) (x e^x ) d x - Vidyadip

Question

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

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Answer

put $x \cdot e^x=t$
Differentiating both sides
$
\begin{aligned}
& \left(x \cdot e^x+e^x \cdot 1\right) \cdot d x=1 d t \\
& e^x(1+x) \cdot d x=1 d t
\end{aligned}
$$\begin{aligned} \mathrm{I} & =\int \frac{1}{\cos t} \cdot d t \\ & =\int \sec t \cdot d t \\ & =\log (\sec t+\tan t)+c \\ & =\log \left(\sec \left(x e^x\right)+\tan \left(x e^r\right)\right)+c\end{aligned}$

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