Mathematics

# Evaluate the following integral:$\displaystyle \int { \cfrac { \left( x+1 \right) { e }^{ x } }{ \sin ^{ 2 }{ \left( x{ e }^{ x } \right) } } } dx$

##### SOLUTION
$\displaystyle\int{\dfrac{\left(x+1\right){e}^{x}dx}{{\sin}^{2}{\left(x{e}^{x}\right)}}}$

Let $t=x{e}^{x}$

$\Rightarrow\,dt=\left(x{e}^{x}+{e}^{x}\right)dx=\left(x+1\right){e}^{x}dx$

$=\displaystyle\int{\dfrac{dt}{{\sin}^{2}{t}}}$

$=\displaystyle\int{{\csc}^{2}{t}\,dt}$

$=-\cot{t}+c$

$=-\cot{\left(x{e}^{x}\right)}+c$ where $t=x{e}^{x}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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