Proof of Integration by PartsDate: 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/ |
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