State and prove second mean-value theorem for Riemann-integrals?

The Second Mean Value Theorem for Riemann integrals states that if ( f ) and ( g ) are continuous functions on the closed interval ([a, b]) and ( g ) is non-negative and integrable, then there exists a point ( c \in [a, b] ) such that:

[ \int_a^b f(x) g(x) , dx = f(c) \int_a^b g(x) , dx. ]

Proof: Define ( G(x) = \int_a^x g(t) , dt ). Since ( g ) is continuous, ( G ) is differentiable and ( G(a) = 0 ). By applying the Mean Value Theorem to ( G ) over ([a, b]), we find a ( c \in [a, b] ) such that:

[ G(b) = G'(c)(b - a) = g(c)(b - a). ]

Thus, we have:

[ \int_a^b g(x) , dx = G(b) = g(c)(b - a), ]

which leads to the conclusion that:

[ \int_a^b f(x) g(x) , dx = f(c) \int_a^b g(x) , dx. ]