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
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