To solve an equation

I am trying to solve the equation:

Solve[Re[Integrate[(2 c - (r^2) Exp[- I 1.4746])^(0.5), {r, 0, 31.7576}]] == 2.1216, c]


I want to integrate an expression in the unknown $c$, in some integration interval, then evaluate the real part, and at last solve the equation, and obtain $c$. I don't know why nothing has been returned.

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Something like this f[c_?NumericQ] := Re[NIntegrate[(2 c - (r^2) Exp[-I 1.4746])^(0.5), {r, 0, 31.7576}]] - 2.1216 FindRoot[f[c], {c, 1}] gives an answer. Probably best to think more about the mathematical problem first, though. – acl Jun 30 '14 at 15:53
is c>0? if so, int = Assuming[c > 0, Integrate[(2 c - (r^2) Exp[-I 1.4746])^(0.5), {r, 0, 31.7576}]] gives the answer in Hypergeometric2F1 – Nasser Jun 30 '14 at 15:58
I should obtain c. But, I expect to be positive. – Otto Jun 30 '14 at 16:01
Plot the function and see why you get what you get... – acl Jun 30 '14 at 16:53
I got c=-2.89464 E6. – Algohi Jun 30 '14 at 18:18