Correct option is (4) I(9, 13)
\(I(m, m)=\int_{0}^{1} x^{m-1}(1-x)^{n-1} d x\)
Let \(x=\sin ^{2} \theta \quad d x=2 \sin \theta \cos \theta d \theta\)
\(I(m, n)=2 \int_{0}^{\pi / 2}(\sin \theta)^{2 m-1}(\cos \theta)^{2 n-1} d \theta\)
\(I(9,14)+I(10,13)=2 \int_{0}^{\pi / 2}(\sin \theta)^{17}(\cos \theta)^{27} d \theta\)
\(+2 \int_{0}^{\pi / 2}(\sin \theta)^{19}(\cos \theta)^{25} \mathrm{~d} \theta\)
\(=2 \int_{0}^{\pi / 2}(\sin \theta)^{17}(\cos \theta)^{25} d \theta\)
= I(9, 13)
