Mathematics

If $$\displaystyle \int \frac{dx}{\left ( x-p \right )\sqrt{\left ( x-p \right )\left ( x-q \right )}} \displaystyle =-\frac{2}{p-q}\sqrt{\frac{x-a}{x-b}}+c$$ then find $$a$$ and $$b$$ are respectively


ANSWER

$$q,p$$


SOLUTION
Let $$\displaystyle \sqrt{\frac{x-q}{x-p}}=t$$

Differentiate on both sides

$$\displaystyle \therefore \frac{1}{2}\left ( \frac{x-p}{x-q} \right )^{\tfrac12}\frac{\left ( x-p \right )1-\left ( x-q \right )1}{\left (x-p  \right )^{2}}dx=dt.$$

$$\displaystyle \Rightarrow \frac{1}{2}\frac{q-p}{\sqrt{x-q\left ( x-p \right )^{3}}}dx=dt$$

$$\displaystyle \Rightarrow \frac{dx}{\left ( x-p \right )^{\tfrac 12}\sqrt{\left ( x-q \right )(x-p)}}=\frac{2dt}{q-p}$$

$$\displaystyle \Rightarrow \:I=-\int \frac{2dt}{p-q}=\dfrac{-2}{p-q}t+c$$

$$\displaystyle =-\frac{2}{p-q}\sqrt{\frac{x-q}{x-p}}+c$$

by comparing we get $$\boxed{a=q,b=p\ }$$
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Single Correct Hard Published on 17th 09, 2020
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