Nail-Tinted Glasses

Differentiating under the integral sign

I used that one damn tool again and again. -- Feynman.

Now that I have Advanced Calculus, by F. S. Woods, and also Higher Mathematics for engineers and Physicists by Sokolnikoff and Sokolnikoff, it's time to write the webpage that I could never find.

What is "differentiation under the integral sign", how does it work, and most importantly what sorts of problems bring it into play?

Differentiation under the integral sign - referenced in Woods as "Differentiation of a definite integral", is the application of the Leibniz Integral Rule to solve an integral.

The general recipe is that you take an integral in the first variable which is hard, take the derivative with respect to a second variable underneath the integral sign to produce a simpler integral, compute this integral (of the first variable), then take an integral of the second variable to finish up.

With that general idea, it would be nice to have some practice problems. I eventually intend to annotate these with solutions....

∫ ln (1-2acos(x)+a^2) dx from 0 to π [Woods]

∫ ln (1+acos(x)) dx from 0 to π [Sokolnikoff]

∫ dx / ( a - cos(x))^2 from 0 to π [Sokolnikoff]

March 11, 2004 11:07 PM | TrackBack