I am interested in the evaluation of
$\int^1_0 \left(1-x^2\right)^{n/2} \sin(kx)\, dx$
Really $n$ is an odd integer here (the even case is easy).
Ooops, I think the posting
Integral $\int_{-1}^1 \frac{e^x}{\sqrt{1-x^2}}dx$
pretty much covers it, esp. the answer from J.M.