# Why can't Mathematica solve this integral in definite form?

Posted 9 years ago
10243 Views
|
3 Replies
|
3 Total Likes
|
 Hi guys, Could you help me out? Why can't Mathematica solve this integral in definite form? It's just stuck computing. Integrate[(1/Sqrt[(x - 1) (x^2 - a^2)]) - (1/ Sqrt[(x + 1) (x^2 - a^2)]), {x, a, b}] ->And it stuckIntegrate[(1/Sqrt[(x - 1) (x^2 - a^2)]) - (1/ Sqrt[(x + 1) (x^2 - a^2)]] -> this gets solvedAm I doing something wrong?Thanks, M.
3 Replies
Sort By:
Posted 9 years ago
 Even if you use Assumptions->1 b > a > 1] (2*((2*I)*Sqrt[(-1 + a^2)*(-1 + b)] - (2*I)*Sqrt[(-1 + a^2)*(1 + b)] + Sqrt[(1 + a)*(-1 + b^2)]*EllipticF[ArcSin[Sqrt[(a + b)/a]/Sqrt[2]], (2*a)/(-1 + a)] - Sqrt[(-1 + a)*(-1 + b^2)]* EllipticF[ArcSin[Sqrt[(a + b)/a]/Sqrt[2]], (2*a)/(1 + a)] - Sqrt[(1 + a)*(-1 + b^2)]*EllipticK[(2*a)/(-1 + a)] + Sqrt[(-1 + a)*(-1 + b^2)]*EllipticK[(2*a)/(1 + a)]))/ Sqrt[(-1 + a^2)*(-1 + b^2)] 
Posted 9 years ago
 Wow thanks, Marco. Thank you for the detailed answer. This does indeed seem to be the problem (I should have checked that myself).
Posted 9 years ago
 The problem might be the singularity at the lower integration boundary. When x equals a the denominators become zero. You can also see that when you plot the integrand for some values a and b: Plot[(1/Sqrt[(x - 1) (x^2 - a^2)]) - (1/Sqrt[(x + 1) (x^2 - a^2)]) /. a -> 1, {x, 1, 2}] You can also check that with Wolfram alpha:WolframAlpha["Domain of (1/Sqrt[(x - 1) (x^2 - a^2)]) - (1/Sqrt[(x + \ 1) (x^2 - a^2)])", {{"Result", 1}, "Output"}]This gives:HoldComplete[(a < -1 && (-Sqrt[a^2] < x < -1 || x > Sqrt[a^2])) || (-1 <= a <= 1 && x > 1) || (a > 1 && (-Sqrt[a^2] < x < -1 || x > Sqrt[a^2]))]When you solve the indefinite integral: Integrate[(1/Sqrt[(x - 1) (x^2 - a^2)]) - 1/Sqrt[(x + 1) (x^2 - a^2)], x] you obtain (2 I Sqrt[1 - (-1 + a)/(-1 + x)] Sqrt[1 + (1 + a)/(-1 + x)] (-1 + x)^(3/2) Sqrt[-a^2 + x^2] EllipticF[I ArcSinh[Sqrt[1 + a]/Sqrt[-1 + x]](1 - a)/(1 + a)])/(Sqrt[1 + a] Sqrt[1 - a^2 + 2 (-1 + x) + (-1 + x)^2]Sqrt[(-1 + x) (-a^2 + x^2)]) - (2 I (1 + x)^(3/2) Sqrt[-a^2 + x^2] Sqrt[1 + (-1 + a)/(1 + x)] Sqrt[1 - (1 + a)/(1 + x)] EllipticF[I ArcSinh[Sqrt[-1 - a]/Sqrt[1 + x]], (1 - a)/(1 + a)])/(Sqrt[-1 - a] Sqrt[(1 + x) (-a^2 + x^2)] Sqrt[1 - a^2 - 2 (1 + x) + (1 + x)^2]) `which is also clearly not defined at x==1. Cheers, Marco