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Question Number 95198 by Fikret last updated on 23/May/20

$$\int \sqrt{\frac{x+2}{x-3}}dx=?$$

Answered by MJS last updated on 23/May/20

$$\int \sqrt{\frac{x+2}{x-3}}dx=$$$$[t=\sqrt{\frac{x+2}{x-3}}\to dx=-\frac{2}{5}\sqrt{{(x-3)}^3(x+2)}]$$$$=-10\int \frac{t^2}{{(t^2-1)}^2}dt=$$$$=-\frac{5}{2}(\int \frac{dt}{{(t-1)}^2}+\int \frac{dt}{{(t+1)}^2}+\int \frac{dt}{t-1}-\int \frac{dt}{t+1})=$$$$=\frac{5}{2}(\frac{1}{t-1}+\frac{1}{t+1}-\ln (t-1)+\ln (t+1))=$$$$=\frac{5}{t^2-1}+\frac{5}{2}\ln \frac{t+1}{t-1}=$$$$=\sqrt{(x-3)(x+2)}+\frac{5}{2}\ln (2x-1+2\sqrt{(x-3)(x+2)})+C$$

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