what is the anti-derivative of cos^5x

Answer 1

cos^5θ = cos^4θ cosθ dθ
= (1 - sin^2θ)^2 cosθ dθ
= 1 - 2sin^2θ + sin^4θ cosθ dθ

Now, note that if u = sinθ then du = cosθ dθ and we have

1 - 2u^2 + u^4 du

Easy peasie lemon-squeezy:

u - 2/3 u^3 + 1/5 u^5 + C
sinθ - 2/3 sin^3θ + 1/5 sin^5θ + C