*“I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. [It] showed how to differentiate parameters under the integral sign — it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. [If] guys at MIT or Princeton had trouble doing a certain integral, [then] I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.” (Richard P. Feynman, “Surely You’re Joking, Mr. Feynman!”, Bantam Book, 1985)*

“Feynman’s Trick” is a powerful technique for evaluating nontrivial definite integrals. It is based on Leibniz’s rule (**LR-1**) which states:

Let be a differentiable function in with continuous. Then

This is how it works in practice:

To evaluate definite integral

we introduce into integrand a parameter such that

when

and

when

Suppose

By Leibniz’s rule,

Integrate (4) with respect to :

where

Let

Let

And so,

Now, let’s play “Feynman’s Trick” on definite integral

Differentiate with respect to we have

It means

Hence,

Let ,

*Exercise-1* Given where Show that