\[\int_{0}^{1}(1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda)}_{2n+1}\left(x\right)%
\sin\left(xy\right)\,\mathrm{d}x=\frac{(-1)^{n}\pi\Gamma\left(2n+2\lambda+1%
\right)J_{2n+\lambda+1}\left(y\right)}{(2n+1)!\Gamma\left(\lambda\right)(2y)^{%
\lambda}}.\]