Question
what is the anti-derivative of cos^5x
Answers
Steve
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
= (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