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Question Number 40008 by math khazana by abdo last updated on 15/Jul/18

$$1)find\int \frac{dx}{(x+1)\sqrt{x}+x\sqrt{x+1}}.$$

Commented by abdo mathsup 649 cc last updated on 15/Jul/18

$$letI=\int \frac{dx}{(x+1)\sqrt{x}+x\sqrt{x+1}}wehave$$$$I=\int \frac{(x+1)\sqrt{x}-x\sqrt{x+1}}{{(x+1)}^{2}x-x^2(x+1)}dx$$$$=\int \frac{(x+1)\sqrt{x}-x\sqrt{x+1}}{(x+1)x(x+1-x)}dx$$$$=\int \frac{(x+1)\sqrt{x}-x\sqrt{x+1}}{x(x+1)}dx$$$$=\int \frac{dx}{\sqrt{x}}-\int \frac{dx}{\sqrt{x+1}}$$$$=2\sqrt{x}-2\sqrt{x+1}+c.$$

Answered by ajfour last updated on 15/Jul/18

$$I=\int \frac{[(x+1)\sqrt{x}-x\sqrt{x+1}]dx}{x(x+1)[x+1-x]}$$$$=\int \frac{dx}{\sqrt{x}}-\int \frac{dx}{\sqrt{x+1}}$$$$=2\sqrt{x}-2\sqrt{x+1}+c.$$

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