Proof of Integration by Parts

Date: 02/06/2001 at 14:31:20
From: Kamrul
Subject: Integration of uv

Can you please tell me the integration formula for u(x)v(x)?

Date: 02/06/2001 at 21:20:32
From: Doctor Jordi
Subject: Re: Integration of uv

Hello, Kamrul - thanks for writing to Dr. Math.

I imagine you are talking about a procedure called integration by 
parts. It really is nothing more than a by-product of the product 
rule (no pun intended) for differentiation. Check it out:

     d(f(x)*g(x))/dx = f(x)*g'(x) + f'(x)*g(x)

We solve for f(x)*g'(x):

     f(x)*g'(x) = d(f(x)*g(x))/dx - f'(x)*g(x)

Integrate both sides and recall that an integral is the 
antiderivative:

     INT[f(x) g'(x)dx] = f(x)*g(x) - INT[ f'(x) g(x) dx]

Then change to the common notation:

     u  = f(x)
     v  = g(x)
     du = f'(x)dx
     dv = g'(x)dx

This gives us the formula for integration by parts:

     INT[u dv] = uv - INT[v du]

I hope that helped. Write back if you have more questions.

- Doctor Jordi, The Math Forum
  http://mathforum.org/dr.math/