Definite Integral from -a to a of Power of a plus x by Power of a minus x

Theorem

 * $\displaystyle \int_{-a}^a \paren {a + x}^{m - 1} \paren {a - x}^{n - 1} \rd x = \paren {2 a}^{m + n - 1} \frac {\map \Gamma m \map \Gamma n} {\map \Gamma {m + n} }$

where:
 * $\Gamma$ denotes the Gamma function
 * $a$, $m$ and $n$ are positive real numbers.

Proof
Note the resemblance of this result to the integral defining the beta function.

In view of this, we apply the substitution:


 * $\displaystyle u = \frac {a + x} {2 a}$

We then have, by Derivative of Power:


 * $\displaystyle \frac {\d u} {\d x} = \frac 1 {2 a}$

and:

So that:


 * $a + x = 2 a u$

and:


 * $a - x = \paren {2 a} \paren {1 - u}$

and:


 * $\displaystyle \frac {\d x} {\d u} = 2 a$

We therefore have: