Mathematics

# Solve: $\displaystyle \int^{a}_{0} \dfrac{dx}{\sqrt{ax - x^2}}$

##### SOLUTION
$\displaystyle\int^a_0\dfrac{dx}{\sqrt{ax-x^2}}$
$=\displaystyle\int^a_0\dfrac{dx}{\sqrt{\dfrac{a^2}{4}-\dfrac{a^2}{4}+2\times \dfrac{a}{2}x-x^2}}$
$=\displaystyle\int^a_0\dfrac{dx}{\sqrt{\dfrac{a^2}{4}-(x-\dfrac{a}{2})^2}}$
$=\sin^{-1}\left(\dfrac{x-\dfrac{a}{2}}{\dfrac{a}{2}}\right)\displaystyle\int^a_0$
$=\sin^{-1}\left(\dfrac{a-\dfrac{a}{2}}{\dfrac{a}{2}}\right)-\sin^{-1}\left(\dfrac{0-\dfrac{a}{2}}{\dfrac{a}{2}}\right)$
$=\dfrac{\pi}{2}-\left(-\dfrac{\pi}{2}\right)$
$=\pi$.

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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