Evaluate ∫ cos^5 x dx, x ∈ [0, π]

Question

Evaluate \(\int\limits_{0}^{\pi} \) cos⁵ x dx

Answer 1

Let

I = \(\int\limits_{0}^{\pi} \) cos⁵ x dx

Consider cos⁵x= cos⁴x × cosx

= (cos²x)² × cosx

= (1- sin²x)²cos

 Let sinx=y ⇒ cosxdx=dy (Differentiating both sides)

Also, when x=0, y=0 and x=π, y=0

Hence, I become