Integration using parametric derivatives

In calculus, integration by parametric derivatives, also called parametric integration,[1] is a method of integrating certain functions. It is often used in Physics, and is similar to integration by substitution.


For example, suppose we want to find the integral

Since this is a product of two functions that are simple to integrate separately, repeated integration by parts is certainly one way to evaluate it. However, we may also evaluate this by starting with a simpler integral and an added parameter, which in this case is t = 3:

This converges only for t > 0, which is true of the desired integral. Now that we know

we can differentiate both sides twice with respect to t (not x) in order to add the factor of x2 in the original integral.

This is the same form as the desired integral, where t = 3. Substituting that into the above equation gives the value:


  1. Zatja, Aurel J. (December 1989). "Parametric Integration Techniques | Mathematical Association of America" (PDF). Mathematics Magazine. Retrieved 23 July 2019.

WikiBooks: Parametric_Integration

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.