# Reduction formual for integral

#### sigma1

Hello can anyone point me to some good sources for taking the reduction formula for an integral. my text does not carry this topic.

#### AllanCuz

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 $$\displaystyle V = -cos(x)$$

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

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

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

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.