Integration using parametric derivatives
In calculus, integration by parametric derivatives, also called parametric integration, 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:
- Zatja, Aurel J. (December 1989). "Parametric Integration Techniques | Mathematical Association of America" (PDF). www.maa.org. Mathematics Magazine. Retrieved 23 July 2019.