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Question Number 26395 by abdo imad last updated on 25/Dec/17

$$find\int \frac{dx}{x\sqrt{x^2+x-1}}$$

Answered by $@ty@m last updated on 25/Dec/17

$$Letx=\frac{1}{t}$$$$dx=\frac{-1}{t^2}dt$$$$\int \frac{\frac{-1}{t^2}dt}{\frac{1}{t}\sqrt{\frac{1}{t^2}+\frac{1}{t}-1}}$$$$\int \frac{\frac{-1}{t^2}dt}{\frac{1}{t^2}\sqrt{1+t-t^2}}$$$$=\int \frac{-dt}{\sqrt{1-\frac{1}{4}+\frac{1}{4}+t-t^2}}$$$$=\int \frac{-dt}{\sqrt{{\left(\frac{\sqrt{3}}{2}\right)}^2+{(\frac{1}{2}-t)}^2}}$$$$=\ln \mid (\frac{1}{2}-t)+\sqrt{1+t-t^2}\mid +C$$$$=\ln \mid (\frac{1}{2}-\frac{1}{x})+\sqrt{1+\frac{1}{x}-\frac{1}{x^2}}\mid +C$$

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