Mathematics

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


SOLUTION
Let 

$$t={\sin}^{-1}{x}\Rightarrow\,dt=\dfrac{dx}{\sqrt{1-{x}^{2}}}$$

$$\displaystyle\int{\dfrac{{\left({e}^{{\sin}^{-1}{x}}\right)}^{2}}{\sqrt{1-{x}^{2}}}dx}$$

$$=\displaystyle\int{{e}^{2t}dt}$$

$$=\dfrac{{e}^{2t}}{2}+c$$

$$=\dfrac{1}{2}{\left({e}^{{\sin}^{-1}{x}}\right)}^{2}+c$$ where $$t={\sin}^{-1}{x}$$
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Subjective Medium Published on 17th 09, 2020
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