Intro to Parametric Integration

Parametric integration is one such technique that once you are made
aware of it, you will never for the love of god forget it. It goes by many names : ‘Differentiation under the Integral sign’, ‘Feynman’s famous trick’ , ‘Parametric Integration’ and so on.

Let me
demonstrate :

Now this integral might seem familiar to you if you have taken a calculus course before and to evaluate it is rather simple as well.

image

Knowing this you can do lots of crazy stuff. Lets differentiate this
expression wrt to the parameter in the integral – s (Hence the name
parametric integration ). i.e

image
image

Look at that, by simple differentiation we have obtained the expression
for another integral. How cool is that! It gets even better.


Lets differentiate it once more:

image
image

.

.

.

If you keep on differentiating the expression n times, one gets this :

image

Now substituting the value of s to be 1, we obtain the following
integral expression for the factorial. This is known as the gamma
function.

image

There are lots of ways to derive the above expression for the gamma
function, but parametric integration is in my opinion the most subtle
way to arrive at it. 😀

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