Theorem (integration by parts)
Let f and g be function differentiable on an interval I. Then fg′ is integrable on I if and only if f ′g is integrable on I.
Moreover, if F is an antiderivative of f ′g on I, then fg − F is an antiderivative of fg′ on I.

Practical vesion:
Let f and g be function differentiable on an interval I. Then fg′ is integrable on I if and only if f ′g is integrable on I, and