In $\mathrm{I}(\mathrm{m},\mathrm{n})={\int }_0^1 {\mathrm{x}}^{\mathrm{m}-1}(1-\mathrm{x}{)}^{\mathrm{n}-1}\mathrm{dx},\mathrm{m},\mathrm{n}>0$, then I(9, 14) + I(10, 13) is 

$\mathrm{I}(\mathrm{m},\mathrm{m})={\int }_0^1 {\mathrm{x}}^{\mathrm{m}-1}(1-\mathrm{x}{)}^{\mathrm{n}-1}\mathrm{dx}$
Let $\mathrm{x}={\sin }^2\mathrm{\theta }\mathrm{dx}=2\sin \mathrm{\theta cos}\mathrm{\theta d\theta }$
$\begin{array}{l}\mathrm{I}(\mathrm{m},\mathrm{n})=2{\int }_0^{\mathrm{\pi }/2} (\sin \mathrm{\theta }{)}^{2\mathrm{m}-1}(\cos \mathrm{\theta }{)}^{2\mathrm{n}-1}\mathrm{d\theta }\\ \mathrm{I}(9,14)+\mathrm{I}(10,13)=2{\int }_0^{\mathrm{\pi }/2} (\sin \mathrm{\theta }{)}^{17}(\cos \mathrm{\theta }{)}^{27}\mathrm{d\theta }+2{\int }_0^{\mathrm{\pi }/2} (\sin \mathrm{\theta }{)}^{19}(\cos \mathrm{\theta }{)}^{25}\mathrm{d\theta }\\ =2{\int }_0^{\mathrm{\pi }/2} (\sin \mathrm{\theta }{)}^{17}(\cos \mathrm{\theta }{)}^{25}\mathrm{d\theta }\\ =\mathrm{I}(9,13)\end{array}$