I have this simple integral but I don't know why Mathematica can't solve it?!
FullSimplify[Integrate[x/(-x^2 + a x + b)^(1/2), {x, Sqrt[b], Sqrt[R^2 + b]},Assumptions -> {a > 0, b > 0}]]
Clear["Global`*"]
Include the assumption that R
is real.
Assuming[{a > 0, b > 0, Element[R, Reals]},
Integrate[x/(-x^2 + a x + b)^(1/2), {x, Sqrt[b], Sqrt[R^2 + b]}] //
FullSimplify]
We can test the result by uses Newton-Leibniz formula.
f = Integrate[x/(-x^2 + a x + b)^(1/2), x];
expr2 = (f /. x -> Sqrt[R^2 + b]) - (f /. x -> Sqrt[b])
expr1 = Integrate[
x/(-x^2 + a x + b)^(1/2), {x, Sqrt[b], Sqrt[R^2 + b]},
Assumptions -> {a > 0, b > 0, R > 0}]
FullSimplify[expr1 == expr2, Assumptions -> {a > 0, b > 0, R > 0}]
True