Let I(m,n)=_0^1x^m-1(1-x)^n-1,dx, m,n0.

Question

Let $I(m,n)=\displaystyle\int_0^1x^{m-1}(1-x)^{n-1}\,dx$, $m,n>0$. Then $I(9,14)+I(10,13)$ equals:

Accepted Answer

With $x=\sin^2\theta$: $I(m,n) = 2\displaystyle\int_0^{\pi/2}(\sin\theta)^{2m-1}(\cos\theta)^{2n-1}d\theta$.

$$I(9,14)+I(10,13) = 2\int_0^{\pi/2}(\sin\theta)^{17}(\cos\theta)^{27}d\theta + 2\int_0^{\pi/2}(\sin\theta)^{19}(\cos\theta)^{25}d\theta$$

Combining (using $\sin^2\theta+\cos^2\theta=1$ on the common factor) reduces this to $2\displaystyle\int_0^{\pi/2}(\sin\theta)^{17}(\cos\theta)^{25}d\theta = I(9,13)$.

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