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