Mathematics

Evaluate the following integral
$$\int { \cfrac { 1 }{ \sqrt { 1-{ x }^{ 2 } } \left( 2+3\sin ^{ -1 }{ x }  \right)  }  } dx\quad $$


SOLUTION
Let 
$$t=2+3{\sin}^{-1}{x}$$

$$\Rightarrow\,dt=\dfrac{3}{\sqrt{1-{x}^{2}}}dx$$

$$\Rightarrow\,dt={3}\dfrac{dx}{\sqrt{1-{x}^{2}}}$$

$$\displaystyle\int{\dfrac{dx}{\sqrt{1-{x}^{2}}\left(2+3{\sin}^{-1}{x}\right)}}$$

$$=\dfrac{1}{3}\displaystyle\int{\dfrac{dt}{t}}$$

$$=\dfrac{1}{3}\log{\left|t\right|}+c$$

$$=\dfrac{1}{3}\log{\left|2+3{\sin}^{-1}{x}\right|}+c$$ where $$t=2+3{\sin}^{-1}{x}$$

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Subjective Medium Published on 17th 09, 2020
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