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

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

Answered by MJS_new last updated on 22/Aug/21

$$u^{\prime }=1\to u=x$$$$v=\arcsin \sqrt{\frac{x}{x+a}}\to v^{\prime }=\frac{\sqrt{a}}{2(x+a)\sqrt{x}}$$$$\int \arcsin \sqrt{\frac{x}{x+a}}dx=$$$$=x\arcsin \sqrt{\frac{x}{x+a}}-\frac{\sqrt{a}}{2}\int \frac{\sqrt{x}}{x+a}dx=$$$$=x\arcsin \sqrt{\frac{x}{x+a}}-\sqrt{ax}+a\arcsin \sqrt{\frac{x}{x+a}}=$$$$=-\sqrt{ax}+(x+a)\arcsin \sqrt{\frac{x}{x+a}}+C$$

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