In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, integration

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 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

Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...

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:

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