How is this done: ∫(cos(x))^5 dx?

Question

I'm pretty sure it involves substitution and the fact that (cos(x))^2 = 1 - (sin(x))^2.

Please help me ***understand*** this.

I want to understand how this is done.

Answer 1

∫(cos(x))^5 dx
=∫(cos(x))^4 cosx dx
=∫((cos^2 (x))^2) cosx dx
=∫(1-sin^2 (x))^2 cosx dx
let sinx=u
=∫(1-u^2)^2 du
=∫ (1+u^4-2u^2)du
=u+u^5/5-2u^3/3
=sinx+1/5 * sin^5 (x) -2/3*sin^3 (x) +C

Answer 2

Hey,

Yes, understanding is the key. So here we go

cos(x) ^5 dx = cos(x)^4 cos(x) dx = cos(x)^4 d(sin(x))

Right? since d sin(x) = cos(x) dx

The equation above in turn is nothing else but (as you mentioned)

(1-sin(x)^2)^2 d(sin(x)) which is the same as (1 - y^2)^2 dy which is trivially integrated now since this is a polynomial

The answer is const + y -2/3 *y^3 + 1/5 y^5 where y = sin(x)

Answer 3

You are on the right track. Rewrite the original integral as ((cos(x))^2)^2 * cos(x)
then use your trig identity to replace the (cos(x))^2

Answer 4

back up in ur study of calculas with a minor in algebraics

Answer 5

what you do is go buy a scientific calculator and let it do the work for you