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
Questions 203525
Subjects 9
Chapters 126
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