## Theorem 7.2.5: Integration by Parts

Suppose

*f*and*g*are two continuously differentiable functions. Let*G(x) = f(x) g(x)*. Thenf(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:

Therefore the functionG(x) = [ f(x) g(x) ] = f'(x) g(x) + f(x) g'(x)

*G*is an antiderivative of the function

*f'(x) g(x) + f(x) g'(x)*which means that

But that is equivalent to the statement we want to prove.G(b) - G(a) = f'(x) g(x) + f(x) g'(x) dx =

= f'(x) g(x) dx + f(x) g'(x) dx