sin-1-x-a-x-dx-

Question Number 151587 by peter frank last updated on 22/Aug/21

\int \sin^{- 1} \sqrt{\frac{x}{a + x}} dx

Answered by MJS_new last updated on 22/Aug/21

u^{'} = 1 \to u = x

v = \arcsin \sqrt{\frac{x}{x + a}} \to v^{'} = \frac{\sqrt{a}}{2 (x + a) \sqrt{x}}

\int \arcsin \sqrt{\frac{x}{x + a}} d x =

= x \arcsin \sqrt{\frac{x}{x + a}} - \frac{\sqrt{a}}{2} \int \frac{\sqrt{x}}{x + a} d x =

= x \arcsin \sqrt{\frac{x}{x + a}} - \sqrt{a x} + a \arcsin \sqrt{\frac{x}{x + a}} =

= - \sqrt{a x} + (x + a) \arcsin \sqrt{\frac{x}{x + a}} + C

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