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First , i multiply numerator and denominator by (2-x-x^2)^(1/2),

then I split the integral into 2 parts ,

using trig substitution , part 2 is easy to be integrated ..

but when I tried to integrate part 2 , I was completely stuck, I tried trig substitution and integration by parts but none worked

Can anyone help me ?

user143997
  • 77
  • I use Maple to help me with the integral and the answer contains hyperbolic tangent inverse functions. – Novice Jan 05 '15 at 08:02
  • Try setting $\sin\alpha = \frac23(x+\frac12)$ as you did, but then do the "universal trig substitution" $y=\tan\frac\alpha2$; the result will be an integral of a rational function, which you can do using partial fractions. I think it will be a little easier if you do integration by parts as the very first step, integrating $\frac1{x^2}$ and differentiating $\sqrt{2-x-x^2}$, and then doing these substitutions. – Greg Martin Jan 05 '15 at 09:07
  • oh ！ I see ,great !thank you so much , Greg I was misguided by the hint on the textbook – user143997 Jan 05 '15 at 09:16
  • See Euler substitutions. – David H Jan 05 '15 at 13:47

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