find-dx-x-x-2-x-1-

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

f i n d \int \frac{d x}{x \sqrt{x^{2} + x - 1}}

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

L e t x = \frac{1}{t}

d x = \frac{- 1}{t^{2}} d t

\int \frac{\frac{- 1}{t^{2}} d t}{\frac{1}{t} \sqrt{\frac{1}{t^{2}} + \frac{1}{t} - 1}}

\int \frac{\frac{- 1}{t^{2}} d t}{\frac{1}{t^{2}} \sqrt{1 + t - t^{2}}}

= \int \frac{- d t}{\sqrt{1 - \frac{1}{4} + \frac{1}{4} + t - t^{2}}}

= \int \frac{- d t}{\sqrt{{(\frac{\sqrt{3}}{2})}^{2} + {(\frac{1}{2} - t)}^{2}}}

= \ln ∣ (\frac{1}{2} - t) + \sqrt{1 + t - t^{2}} ∣ + C

= \ln ∣ (\frac{1}{2} - \frac{1}{x}) + \sqrt{1 + \frac{1}{x} - \frac{1}{x^{2}}} ∣ + C

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