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Question Number 20244 by tammi last updated on 24/Aug/17

$$\int \frac{dx}{(2-x)\sqrt{x}}$$

Answered by ajfour last updated on 24/Aug/17

$$let\sqrt{x}=t\Rightarrow dx=2tdt$$$$\int \frac{dx}{(2-x)\sqrt{x}}=\int \frac{2tdt}{(2-t^2)t}$$$$=-2\int \frac{dt}{t^2-2}=\frac{-2}{2\sqrt{2}}\ln \mid \frac{t-\sqrt{2}}{t+\sqrt{2}}\mid +C$$$$=\frac{1}{\sqrt{2}}\ln \mid \frac{\sqrt{x}+\sqrt{2}}{\sqrt{x}-\sqrt{2}}\mid +C.$$

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