Integration using parametric derivatives

In calculus, integration by parametric derivatives, also called parametric integration,[1] is a method which uses known Integrals to integrate derived functions. It is often used in Physics, and is similar to integration by substitution.

Statement of the theorem

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By using the Leibniz integral rule with the upper and lower bounds fixed we get that
 
It is also true for non-finite bounds.

Examples

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Example One: Exponential Integral

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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:

 

Example Two: Gaussian Integral

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Starting with the integral  , taking the derivative with respect to t on both sides yields
 .
In general, taking the n-th derivative with respect to t gives us
 .

Example Three: A Polynomial

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Using the classical   and taking the derivative with respect to t we get
 .

Example Four: Sums

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The method can also be applied to sums, as exemplified below.
Use the Weierstrass factorization of the sinh function:
 .
Take the logarithm:
 .
Derive with respect to z:
 .
Let  :
 .

References

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