Question

Integration:

Answer 1

\[\LARGE \int_{0}^{\pi}~cos^4(2t)~dt\]

Answer 2

using integral f(x) dx from 0 to t = integral f(t-x) dx from 0 to t,
and also if f(x) = f(t-x) , then integral f(x) dx from 0 to t= 2 integral f(x) dx from 0 to t/2
We get
2I =4 (integral cos^4(2t) + sin^4(2t) ) from 0 to pi/4
Now ,
cos^4(2t) + sin^4(2t) = 1 - 2sin^2(2t)cos^2(2t)
= 1 - sin^2 (4t) /2

We may use sin^2(4t) = (1- cos(8t))/2 and proceed further.