Mathematics

Solve $$\int {\dfrac{dx}{\sqrt{x-x^2}}dx}$$ .


ANSWER

$$ I={{\sin }^{-1}}\left( 2x-1 \right)+C $$


SOLUTION

Consider the given integral.

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

$$ I=\int{\dfrac{dx}{\sqrt{{{\left( \dfrac{1}{2} \right)}^{2}}-{{\left( \dfrac{1}{2} \right)}^{2}}+x-{{x}^{2}}}}} $$

$$ I=\int{\dfrac{dx}{\sqrt{{{\left( \dfrac{1}{2} \right)}^{2}}-{{\left( x-\dfrac{1}{2} \right)}^{2}}}}} $$

We know that

$$\int{\dfrac{dx}{\sqrt{{{a}^{2}}-{{x}^{2}}}}={{\sin }^{-1}}\left( \dfrac{x}{a} \right)}+C$$

Therefore,

$$ I={{\sin }^{-1}}\left( \dfrac{x-\dfrac{1}{2}}{\dfrac{1}{2}} \right)+C $$

$$ I={{\sin }^{-1}}\left( 2x-1 \right)+C $$

Hence, this is the answer.

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Single Correct Medium Published on 17th 09, 2020
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