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

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007