Mathematics

Solve $$\displaystyle \int\sqrt{\dfrac{a-x}{a+x}}dx$$


SOLUTION
$$= \displaystyle\int \dfrac{a-x}{\sqrt{a^2-x^2}}dx $$

$$= \displaystyle\int \dfrac{a}{\sqrt{a^2-x^2}}dx - \int \dfrac{1}{2}\dfrac{2x}{\sqrt{a^2-x^2}}dx$$


$$ let a^2 - x^2 = t$$
$$  -2x = dt$$

$$= a\displaystyle\int \dfrac{1}{\sqrt{a^2-x^2}}dx + \int \dfrac{1}{2}\dfrac{dt}{\sqrt{t}}$$

$$= a\sin^{-1}\dfrac{x}{a}+\dfrac{1}{2}*\dfrac{2}{1}t^{(\dfrac{1}{2})}$$

$$= a\sin^{-1}\dfrac{x}{a}+(a^2-x^2)^{(\dfrac{1}{2})}$$ + $$C $$



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Subjective Medium Published on 17th 09, 2020
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