Interesting proof of an integration formula and some applications

Question

Evaluate

Solution

Let

\ ydy = x dx

\

Some Applications

(a)

, by integration by parts.

By (1), we get:

(b)

Rationalize the denominator of the integrand, we get:

Using (2) and putting a = 1, we get:

Challenge

(1) Prove (2) by trigonometric substitution.

(2) Prove (3) by using the substitution:

and write x in terms of t.