By using the properties of definite integrals, evaluate the integrals`int_0^(2pi)cos^5x dx`

Explanation

Let `I = int_0^(2pi) cos^5x dx`

We know, `int_0^(2a) f(x) dx= 2 int_0^(a) f(x) dx` when `f(2a-x) dx = f(x)`

Here, `cos^5(2pi-x) = cos^5x`

`:. I = 2 int_0^pi cos^5x dx`

Now, `int_0^(a) f(x) dx=0` when `f(a-x) dx = -f(x)`

Here, `cos^5(pi-x) = - cos^5x`

`:. I = 2 int_0^pi cos^5x dx = 2*0 = 0`

`:. int_0^(2pi) cos^5x dx = 0.`