Integration using parametric derivatives

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In mathematics, integration by parametric derivatives is a method of integrating certain functions.

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: