Mathematics - Definite Integration Question with Solution | TestHub

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