# Technical question on integrals.

1. Sep 3, 2006

### MathematicalPhysicist

i have to show that:
1)$$(-1)^n\int_{-1}^{1}(x^2-1)^ndx=2^{2n+1}(n!)^2/(2n+1)!$$
2) $$\binom{n}{k}=[(n+1)\int_{0}^{1}x^k(1-x)^{n-k}dx]^{-1}$$

for the first part i thought to use newton's binomial, i.e:
$$(1-x^2)^n=\sum_{k=0}^{n}\binom{n}{k}(-x^2)^k$$
but it didn't get me far.
for the second part i dont have a clue, i dont think you can integrate the integral by parts or substitution can you?

2. Sep 3, 2006

### StatusX

Where'd you get stuck after using the binomial theorem? Just integrate term by term.

3. Sep 3, 2006

### MathematicalPhysicist

this is what i got:
2-2n/3+2n(n-1)/10+...+2(-1)^n/(2n+1)
i don't know how to procceed from here.

4. Sep 3, 2006

### tim_lou

yeah, i think number 2 can be worked out using integral by parts... the tabular method.
all except the last term of the resulting series is zero when the bounds are substituted in.