Integration Question

1. Feb 25, 2007

Hyari

1. The problem statement, all variables and given/known data
integral cos^5(9*t) dt

2. Relevant equations
half sets?

3. The attempt at a solution
integral cos(9t)^5 dt

integral cos(9t)^2 * cos(9t)^2 * cos(9t)

cos(9t)^2 = (1/2)*[ 1 + cos(18t) ]

integral (1/2)*[ 1 + cos(18t) ] * (1/2)*[ 1 + cos(18t) ] * cos(9t)

Am I on the right path?

2. Feb 25, 2007

gammamcc

Hint:
cos^2 u = 1-sin^2 u

3. Feb 25, 2007

Hyari

u = cos(9t)
du = -9sin(9t)

[ 1 - sin(9t)^2 ] * [ 1 - sin(9t)^2 ] * u du?

4. Feb 26, 2007

theperthvan

$$\cos^5{9t}$$ is the same as $$\cos{9t}(1-\sin^2{9t})^2$$

Expand it and integrate

5. Feb 26, 2007

Hyari

Then how do you integrate that beat .

cos(9t) * (1 - sin(9t)^2)^2 * dt

u = 1-sin(9t)^2
du = -18sin(9t) * cos(9t) * dt

du / -18sin(9t) = cos(9t) * dt

1 / -18sin(9t) <-integral-> u^2 * du

1/-18sin(9t) * u^3

1/-18sin(9t) * (1-sin(9t)^2)^3 ?

6. Feb 26, 2007

gammamcc

try again, try u = something else.

7. Feb 26, 2007

theperthvan

You need to expand it. Then you will be able to use $$\frac{d}{dx}\sin{x} = \cos{x}$$

8. Feb 26, 2007

Gib Z

So basically when we have to integrate something with only cosine in it, and cosine is odd powered, we take the most even powers out and transform those into the sines as mentioned, expand and use substitution u=sin x to do the rest.

9. Feb 26, 2007

Hyari

I don't understand... can you give me an example?

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

I don't understand :(

10. Feb 27, 2007

Gib Z

Do you know the Identity $\sin^2 x + \cos^2 x=1$?

11. Feb 27, 2007

HallsofIvy

Staff Emeritus
What if you let u= sin(9t) instead?