# Another "theoretical" thing about integrals I have no idea how to do

Printable View

• February 6th 2010, 10:48 AM
s3a
Another "theoretical" thing about integrals I have no idea how to do
Question:
If a and b are positive numbers, show that

this(integral (x^a * (1-x)^b) from 0 to 1) = this(http://www.wolframalpha.com/input/?i=integral+(x^b+*+(1-x)^a)+from+0+to+1).

And for this problem, I don't even have the answer in the back of my book! Someone please help!

(Ignore what Wolfram Alpha says - I am just using those links to make it more visual because I don't know how to use latex)

Any help would be GREATLY appreciated!
Thanks in advance!
• February 6th 2010, 11:11 AM
nehme007
Let $u = 1-x$ and $du = -dx$ and you should get:
$\int_0^1 x^a (1-x)^b dx = -\int_1^0 (1-u)^a u^b du = \int_0^1 (1-u)^a u^b du$
And since we're dealing with a definite integral, it doesn't matter that we ended up with u's.
• February 8th 2010, 08:29 AM
s3a
Can you please explain why we assume equality of the variables u and x in the last step? Just stating that we can do so because we're dealing with a definite integral doesn't explain it well enough for me.
• February 8th 2010, 08:41 AM
nehme007
Since it's a definite integral, you ultimately wind up plugging the limits into the antiderivative. So it doesn't matter if you start with x or u, as whichever variable you use will be replaced by the limits 0 and 1 anyway.

If you aren't buying it, just take your favorite definite integral and change the variable. You should get the same answer.