Need help in evaluating $\int_{-1}^1 (1-x^2)^k, k \in \mathbb{N}$ Can someone tell me how to evaluate this integral please?
$$\int_{-1}^1 (1-x^2)^k, k \in \mathbb{N}$$
I tried using the substitution x = sin(t), which would allow me to express this as:
$$\int_{-1}^1 cos^{2k+1}(t) dt$$
but this doesn't really help. Any other tricks?

This is simply a Beta function. Using the substitution $t=x^2$, this reduces to
$$2\int_0^1 \frac{1}{2\sqrt{t}} (1-t)^k\ dt$$
$$\int_0^1 t^{-\frac{1}{2}} (1-t)^k$$
$$B\left(\frac{1}{2},k+1\right)$$
which can be expressed in a lot of different ways (see the linked Wikipedia page). 
