Mark the correct alternative in each of the following:

Question:

Mark the correct alternative in each of the following:

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

A. $2 \log _{e} \cos \left(x e^{x}\right)+C$

B. $\sec \left(x e^{x}\right)+C$

C. $\tan \left(x e^{x}\right)+C$

D. $\tan \left(x+e^{x}\right)+C$

Solution:

let $(t)=x e^{x}$

$\frac{d t}{d x}=e^{x}(1+x)$

$\Rightarrow \int \frac{d t}{(\cos t)^{2}}=\int(\sec t)^{2} d t$

$=\tan t$

(put $\left.(\mathrm{t})=\mathrm{x} e^{x}\right)$

$=\tan \left(\mathrm{x} \mathrm{e}^{\mathrm{x}}\right)+\mathrm{c}$

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