Solving cos^3 x dx: A Quick and Easy Method

[PrudensOptimus]

OK, I know the solution for cos^3 x dx is sinx - sin^3 x / 3 + C.


And that

you basically solve

integral of cosx*(1-sin^2x) dx. to get it.

but,...

what I don't get is how do you solve cosx*(1-sin^x) dx... is there a trick that I didn't get from the parts formula?

 

[Hurkyl]

It would help if you mentioned you're trying to integrate!

Distribute the multiplication and see if that gives you any hints.

 

[HallsofIvy]

Yes, one doesn't normally say "solve f(x)dx"!

Hurkyl, I don't see any reason to "distribute" (multiply out) anything. There is an obvious substitution for ∫(1- sin²(x))cos(x)dx.

 

[Hurkyl]

Good point.

 

[PrudensOptimus]

I have never learned integration by parts. Please help me.

 

[Hurkyl]

But, I presume, you know substitution?

 

[PrudensOptimus]

Nope, any products in Integrals other than those constants are new to me.

 

[Hurkyl]

I didn't ask about products, I asked about substitution!


E.G. would you know how to integrate ∫ sin(πx) dx

 

[PrudensOptimus]

i know the answer,

but I don't know the part when they did the dx = dv(ax) part... that confuzed me.

 

[ShawnD]

Ok well here's how I worked it out

http://myfiles.dyndns.org/pictures/integrate1.jpg

I put a few steps together but you can still see what happened sort of.