Using the Reduction formula

[jaredld]

I am trying to use the reduction formula to express the following integral, but need some help toward the end. Given,\(\displaystyle $\int {\cos ^3 x\;dx = \left[ \matrix{
u = \cos ^2 x \hfill \cr
dv = \cos x\;dx \hfill \cr
du = 2\cos x( - \sin x\;dx) \hfill \cr
v = \sin x \hfill \cr} \right] \to \cos ^2 x\sin x + 2\int {(1 - \cos 2x)\cos x\,dx} } $\)
My Professor then went to the following, but I can't figure out what is happening. Can someone please explain?

\(\displaystyle $\int {\cos ^2 x\sin x + 2\int {\cos x\;dx - 2\int {\cos ^3 x\;dx} } } $\)

 

[ChaoticLlama]

what your professor did was expand the integrand "(1 - cos²(x))cos(x)" into "cos(x) - cos³(x)" so that he could split it up into two separate integrals.

but that is not necessary. Immediately after doing Integration By parts, you are left with:

\(\displaystyle \L\int {\cos ^3 xdx = \sin x\cos ^2 x + 2\int {\sin ^2 x\cos xdx} }\)

The integral that remains in the expression is infact, very easy to integrate, without the use of any identities. Think of chain rule and the answer should come fairly soon.

 

[tkhunny]

You've also a couple of typos in your Parts Expansion. The first integral shouldn't be there and you are missing the exponent on the cosine.

 

[jaredld]

You guys are awsome! thanks for the help. Wooops on the typos... sorry about that. Thanks again everyone! *Corrected*