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I had a problem in the following code because when I use Integrate, there will always be imaginary number in the result. I put my code here. The satisfactory result should look like eq 9.138, which I attached as a photo in this post.

I tried FullSimplify, but it does not work.

Do anyone know how to get rid of the imaginary part and approach eq 9.138?

Remove["Global`*"]

U = k/r;

rmin = r /. Solve[Sqrt[1 - b^2/r^2 - U/T0] == 0, r][[2]]
rmax = Infinity;

alist = {k == Re[k] && Re[k] ≠ 0 && b == Re[b] && Re[b] > 0 && T0 == Re[T0] && Re[T0] > 0};

Θ = Integrate[(b/r) /Sqrt[r^2 - (k/T0) r - b^2], {r, rmin, rmax}, Assumptions -> alist]

Cos[Θ]

enter image description here

enter image description here

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closed as off-topic by Artes, rasher, m_goldberg, belisarius, Verbeia Apr 14 at 23:27

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Try FullSimplify with the same assumptions. –  b.gatessucks Apr 13 at 20:20
1  
That doesn't work here, but ComplexExpand[\[CapitalTheta], TargetFunctions -> {Re, Im}] does. –  Sjoerd C. de Vries Apr 13 at 20:37
    
@b.gatessucks FullSimplify does not work. –  Lawerance Apr 13 at 20:40
1  
Cos[\[CapitalTheta]] // FullSimplify works. –  Artes Apr 13 at 20:44
    
@Artes Yes, it works! –  Lawerance Apr 13 at 20:50