$\int \frac{(1+x) e^x}{\cot \left(x e^x\right)} d x=$
$\int \frac{(1+x) e^x}{\cot \left(x e^x\right)} d x=$
- $\log \left(\cos \left(x e^x\right)\right)+c$
- $\log \left(\cot \left(x e^x\right)\right)+c$
- $\log \left(\sec \left(x e^x\right)\right)+c$
- $\log \left(\operatorname{cosec}\left(x e^x\right)\right)+c$
Solution
$\int \frac{(1+x) \cdot e^x}{\cot \left(x \cdot e^x\right)} d x$
Put, $x e^x=t$
Differentiate w.r.to, $x$
$
\begin{aligned}
\Rightarrow e^x(1+x) d x & =d t \\
& =\int \frac{d t}{\cot t}=\int \tan t d t \\
& =\log (\sec t)+c \\
& =\log \left(\sec \left(x e^x\right)\right)+c
\end{aligned}
$
Hence option (3) is correct
Asked in: AP EAMCET 2020 (22 Sep Shift 2)
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