How Can Reduction Formulas Simplify the Integral of \( x^m(1-x)^n \)?

[kioria]

1. The problem

Show that $$\int^{1}_{0}x^{m}(1-x)^{n} dx = \frac{m!n!}{(m+n+1)!}$$ for all integers $$m, n \geq 0$$

The question is under "Reduction" topic, so I assume we solve this via reduction.

2. My attempt

My attempt is as follows:

Let $$x = cos^{2}x$$

Then we get $$\frac{1}{2}\int cos^{2m-1}sin^{2n-1}dx$$

From here I use the reduction formula: $$I_{m, n} : \frac{m-1}{m+n} : m \geq 2$$ or $$\frac{n-1}{m+n} : n \geq 2$$

It seems like I am on the right track, but it's not working out properly. Am I missing something?

 

[sprint]

don't really know how to help but...

wow. that looks like a monster problem! what hideousness of an equation that is!

 

[chaoseverlasting]

Solve using integration by parts. You should get a redundant term or otherwise be able to simplify the problem in a few steps. The substitution method looks unnecessarily complicated here, and that too is based on integration by parts...so you'd be better off trying this problem from the basics up.