Mathematics

Solve:
$$\displaystyle\int {\dfrac{{3xdx}}{{\sqrt {1 - {x^2}} }}} $$


SOLUTION

Consider the given integral.


$$I=\displaystyle\int{\dfrac{3x}{\sqrt{1-{{x}^{2}}}}}dx$$


 


Let $$t=1-{{x}^{2}}$$


$$ \dfrac{dt}{dx}=0-2x $$


$$ -\dfrac{dt}{2}=xdx $$


 


Therefore,


$$ I=-\dfrac{3}{2}\displaystyle\int{\dfrac{1}{\sqrt{t}}}dt $$


$$ I=-\dfrac{3}{2}\left( 2\sqrt{t} \right)+C $$


$$ I=-3\sqrt{t}+C $$


 


On putting the value of $$t$$, we get


$$I=-3\sqrt{1-{{x}^{2}}}+C$$


 


Hence, this is the answer.

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Subjective Medium Published on 17th 09, 2020
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