Question

Integration question: integral (cos(x))^6

Answer 1

Hmm so we have to do some annoying integration for this one.

After we integrate, we'll be left with an integral involving (cos(x))^4.
So what you'll notice is that we'll effectively decrease the power by 2 each time we integrate.

There is a "Reduction Formula for Cosine" that you can look up in your book probably.
But I'm assuming since it's a homework assignment or whatever, that your teacher wants you to do the steps.

Lemme see if I can remember how to set this up...

Answer 2

Thanks again

Answer 3

I should be allowed to use the half/double angle formulas though.

Answer 4

\[\huge \int\limits \cos^6 dx \quad = \quad \int\limits \color{blue}{\cos^5x}\cdot \color{orangered}{\cos x dx}\]We'll do Integration-by-parts,\[\huge \color{blue}{u=\cos^5x},\qquad \color{orangered}{dv=\cos x dx}\]

Answer 5

Mmmm lemme make sure I didn't make a mistake real quick :D

Answer 6

Seems correct

Answer 7

wait, isn't du -5cos^4(x)*sin(x)?

Answer 8

Ah ty that's where i got messed up :O knew there was suppose to be another power somewhere lol

Answer 9

:)

Answer 10

\[\huge \color{blue}{du=-5\sin x\cdot \cos^4 x dx}, \qquad \color{orangered}{v=\sin x}\]
\[\large =\color{blue}{\cos^5x}\color{orangered}{\sin x}+\color{blue}{5}\int\limits\limits \color{blue}{\sin x\cdot \cos^4 x}\color{orangered}{sinx}\color{blue}{dx}\]

Answer 11

Ignoring the first part for a moment, we have,\[\large 5\int\limits \cos^4x \sin^2x dx \quad = \quad 5\int\limits \cos^4x dx-5 \int\limits \cos^6x dx\]

Answer 12

I skipped a couple steps in there, let me know if it's too confusing.

I changed sin^2 to (1-cos^2), then distributed the cos^4, and wrote it as 2 separate integrals.

Answer 13

yeah its cool

Answer 14

Similar to that annoying problem we did yesterday :) We end up with the same thing we started with!\[\large Y=\cos^5x \sin x+5\int\limits \cos^4x dx-5Y\]

Answer 15

\[\large 6Y=\cos^5x \sin x+5\int\limits\limits \cos^4x dx\]

\[\large Y=\frac{1}{6}\cos^5x \sin x+\frac{5}{6}\int\limits\limits \cos^4x dx\]

Answer 16

From this point, you might want to try and recognize the pattern. And derive the reduction formula from what we're doing. Otherwise doing this 2 more times will be a BEAST!!

Answer 17

lol, Thanks again :)

Answer 18

\[\large \int\limits \cos^n dx=\frac{1}{n}\cos^{n-1}x \sin x+\frac{n-1}{n}\int\limits \cos^{n-2}x dx\]
This is the reduction formula, if you compare it to what we have so far, it will match up.

Answer 19

When you do the next reduction, make sure you multiply all the pieces by that 5/6 fraction.

Then for the final cos^2, I think you had the right idea earlier, Half Angle! :)

Answer 20

all the new pieces*

Answer 21

Thanks I'm going to try that on my test tomorrow :)

Answer 22

Why don't we learn these in class -_-

Answer 23

Hmm good question :D
It's one of those long tedious problems, so it's less likely to show up on a test.
But you know your teacher better than I do :) So I dunno.

Answer 24

yeah, thanks!