Integrating Chebyshev polynomial of the first kind I'm trying to evaluate the integral of the Chebyshev polynomials of the first kind on the interval $-1 \leq x \leq 1 $ .
My idea is to use the closed form 
$$T_n(x) = \frac{z_1^n + z_2^{-n} }{2}$$ 
where $z_1 = (x + \sqrt{x^2 - 1})$ and $z_2 = (x - \sqrt{x^2 - 1})$,
giving the following integral:
$$
\int _{-1}^{1}\!1/2\, \left( x+\sqrt {-1+{x}^{2}} \right) ^{n}+1/2\,
 \left( x-\sqrt {-1+{x}^{2}} \right) ^{n}{}{dx}
$$
I'm stuck at integrating $z_1$ and $z_2$. I tried integrating by parts $n$ times, but i'm looking for a general formula. My calculus is pretty rusty so i'm not sure if this is the way to go. Any tips? 
Thanks a lot.

Wikipedia has a nice article on the Chebychev polynomials: http://en.wikipedia.org/wiki/Chebyshev_polynomials.
In particular, there is this:
$\int T_n(x) dx = \frac{n T_{n+1}(x)}{n^2-1}-\frac{x T_n(x)}{n-1}$.
