# How to integrate cos^3(x)

What is the best way?

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Idk, you didn't show any work.

I guess I'll have to use an identity. Maybe sin^2(x) = 1-cos^2(x)?

(1-cos^2(x))*sin(x)

u = 1- cos^2(x)

du/dx = -2cos(x)sin(x)

so that

(1-cos^2(x))*sin(x) dx = (-u sin(x)/2cos(x)) du

Doesn't really help, or?

$$\int\cos x\cos^2 xdx$$

Use a BASIC trig identity to change the 2nd degree cosine function.

not cosine, sine

Don't know, I guess I'm too drunk to do maths right now.

Don't know, I guess I'm too drunk to do maths right now.
Try again later :)

Oh, I wrote cos^3 x instead of sin^3 x in the headline. That explains my confusion.

sin^3 x

=

sin^2 x*sin x

=

(1 - cos^2 x)sin x

Then substitution?

u = 1-cos^2 x

du/dx = 2cos x*sin*x

so that

sin^3 x dx = - u / 2cos x

Hm...

distribute the sinx and you'll see your solution

HallsofIvy
Homework Helper
Oh, I wrote cos^3 x instead of sin^3 x in the headline. That explains my confusion.

sin^3 x

=

sin^2 x*sin x

=

(1 - cos^2 x)sin x

Then substitution?

u = 1-cos^2 x

du/dx = 2cos x*sin*x

so that

sin^3 x dx = - u / 2cos x

Hm...
So how about just u= cos(x)?

You know what they say "Don't drink and derive"!