#WhyAmIhere's working channel
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$I_1 = \int_0^1 x^m(1-x)^n$
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$I_2= \int_0^1 (1-x)^m x^n$
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$I_1+I_2$
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which is $\int_0^1 x^m(1-x)^n + \int_0^1 (1-x)^m x^n$
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Now to find the bounds of $I(m,n) for (m,n) \eq(2,1))$
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