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wrote...
5 months ago
 What is the integral of √(a-y) /√y What is the integral of √(a-y) /√y Read 68 times 1 Reply
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Educator
5 months ago
 Is it with respect to y or a?Let's assume it's with respect to y, the answer is:$$\sqrt{a}\sqrt{y}\sqrt{1-\frac{y}{a}}+a\arcsin \left(\frac{\sqrt{y}}{\sqrt{a}}\right)$$Here's how, you have to perform trigonometric substitution:First set u = √y$$2\int \sqrt{a-u^2}du$$Then set $$u=\sqrt{a}\sin \left(v\right)\rightarrow v=\sin ^{-1}\left(\frac{u}{\sqrt{a}}\right),\ \frac{du}{dv}=\sqrt{a}\cos \left(v\right)$$Then solve for du.$$=\int \sqrt{a}\cos \left(v\right)\cdot \sqrt{a-a\sin ^2\left(v\right)}dv$$Simplify:$$a-a\sin ^2\left(v\right)=a\cos ^2\left(v\right)$$$$=a\int \cos ^2\left(v\right)dv$$...
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