If g(1)=4, g(5)=5, and ∫ [1,5] g(x) dx =-8, how do you evaluate the integral ∫ [1,5] x g’(x) dx?

Answer 1

∫ x g'(x) dx = ∫ x dg
so, using integration by parts, that is
∫ x dg = (xg - ∫ g dx ) [1..5] = (5g(5) - 1g(1)) - (-8) = 25-4+8 = 29