Something to do with Reduction Formulae...?

Quote:

Originally Posted by j_clough

Hello everybody,

I have got into a terrible mess with a question.

In my question, I am asked to derive the reduction formulae for integrating (sin x)^m * (cos x)^n, which are on the bottom of this webpage:

Reduction Formulas

I keep nearly getting there, but not quite near enough! I think I must be doing the reduction part wrong.

Any help would be greatly appreciated!!

Jessica.

Hello,

What exactly did you do ?

integrate by parts with u=cos^(n-1) (x) and v'=cos(x)*sin^m (x)

this gives :
$I_{m,n}=\frac{\cos^{n-1}(x) \sin^{m+1}(x)}{m+1}+\frac{n-1}{m+1} \int \left(\sin(x) \cos^{n-2}(x)\right)*\left(\sin^{m+1}(x)\right) ~ dx$

$I_{m,n}=\frac{\cos^{n-1}(x) \sin^{m+1}(x)}{m+1}+\frac{n-1}{m+1} \int \cos^{n-2}(x) \sin^m (x) \sin^2(x) ~ dx$

Use the identity $\sin^2(x)=1-\cos^2(x)$ to find $I_{m,n-2}$ and $I_{m,n}$ in the right hand side of the equation =)