Reduction formual for integral

Apr 2010
135
0
Hello can anyone point me to some good sources for taking the reduction formula for an integral. my text does not carry this topic.
 
Apr 2010
384
153
Canada
Hello can anyone point me to some good sources for taking the reduction formula for an integral. my text does not carry this topic.​
"Calculus: a complete course" by Robert A Adams has a very good section on this. I'm not sure about an online resource though...

I'll give you an example of the reduction formula for \(\displaystyle sinx \)

\(\displaystyle \int sin^n x dx = sin^{n-1}(x)sin (x) dx \)

Let \(\displaystyle u= sin^{n-1 }x \) and \(\displaystyle dV = sin(x) dx \)

Such that, \(\displaystyle du = (n-1)sin^{n-2} (x) cos(x) \) and \(\displaystyle V = -cos(x) \)

This leads to,

\(\displaystyle \int sin^n x dx = -sin^{n-1 }x cos(x) + \int (n-1)sin^{n-2} (x) cos^2 (x) dx \)

\(\displaystyle \int sin^n x dx = -sin^{n-1 }x cos(x) +(n-1) \int sin^{n-2} (x) (1-sin^2) dx \)

\(\displaystyle \int sin^n x dx = -sin^{n-1 }x cos(x) + (n-1) \int sin^{n-2} (x) dx - (n-1) \int sin^n dx \)

Notice how \(\displaystyle \int sin^n x dx \) appears on each side so we can expand and factor to get.

\(\displaystyle n \int sin^n x dx = -sin^{n-1 }x cos(x) +(n-1) \int sin^{n-2} (x) \)

\(\displaystyle \int sin^n x dx = -\frac{1}{n} sin^{n-1 }x cos(x) +\frac{(n-1)}{n} \int sin^{n-2} (x) \)

Clearly this is only one example but the reasoning is pretty much the same for all reduction formulas. We identify what will eventually become a re-occuring theme such that we can get it on both the right and left sides by doing by parts a few times.

Most reduction formulas, at least that I know of, are all computed using by parts and it's key we identify which to take as U and which to take as dV. This comes from what I said above and by experience.