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Question Number 27920 by math1967 last updated on 17/Jan/18

$$\int \frac{(x-1)dx}{(x+1)\sqrt{x^3+x^2+x}}$$

Answered by math1967 last updated on 18/Jan/18

$$\int \frac{(x^2-1)dx}{{(x+1)}^2\sqrt{x^3+x^2+x}}$$$$\int \frac{\frac{x^2-1}{x^2}dx}{\frac{(x^2+2x+1)\sqrt{x^3+x^2+x}}{x^2}}$$$$\int \frac{(1-\frac{1}{x^2})dx}{(x+\frac{1}{x}+2)\sqrt{x+\frac{1}{x}+1}}$$$$nowletx+\frac{1}{x}+1=z^2\therefore (1-\frac{1}{x^2})dx=2zdz$$$$\int \frac{2zdz}{(z^2+1).z}=2\int \frac{dz}{z^2+1}={2\tan }^{-1}z+c$$$${2\tan }^{-1}\sqrt{x+\frac{1}{x}+1}+c$$

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