find-ax-x-2-dx-

Question Number 101531 by mhmd last updated on 03/Jul/20

f i n d \int \sqrt{a x - x^{2}} d x

Answered by Mr.D.N. last updated on 03/Jul/20

I = \int \sqrt{ax - x^{2}} dx

= \int \sqrt{- (x^{2} - ax)} dx

= \int \sqrt{{(\frac{a}{2})}^{2} - {(x - \frac{a}{2})}^{2}} dx

= \frac{(x - \frac{a}{2}) \sqrt{{(\frac{a}{2})}^{2} - {(x - \frac{a}{2})}^{2}}}{2} + \frac{{(\frac{a}{2})}^{2}}{2} \sin^{- 1} \frac{(x - \frac{a}{2})}{\frac{a}{2}} + C

= \frac{\frac{2 x - a}{2} \sqrt{ax - x^{2}}}{2} + \frac{a^{2}}{8} \sin^{- 1} \frac{2 x - a}{a} + C

= \frac{(2 x - a) \sqrt{ax - x^{2}}}{4} + \frac{a^{2}}{8} \sin^{- 1} \frac{(2 x - a)}{a} + C / / .

Commented by mhmd last updated on 03/Jul/20

t h a n k y o u s i r

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