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
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