# 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.

## Example

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 *x*^{2} 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:

## References

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