Theorem 7.2.5: Integration by Parts
Suppose f and g are two continuously differentiable
functions. Let G(x) = f(x) g(x). Then
f(x) g'(x) dx = ( G(b) - G(a) ) - f'(x) g(x) dx
Proof:
For the function G(x) = f(x) g(x) we have by the Product Rule:G(x) = [ f(x) g(x) ] = f'(x) g(x) + f(x) g'(x)Therefore the function G is an antiderivative of the function f'(x) g(x) + f(x) g'(x) which means that
G(b) - G(a) = f'(x) g(x) + f(x) g'(x) dx =But that is equivalent to the statement we want to prove.
= f'(x) g(x) dx + f(x) g'(x) dx