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∫(上限1,下限0) x^m *(1-x)^n dx
= 令t=x-(1/2), ∫(上限1/2,下限-1/2) ((1/2)+t)^m *((1/2)-t)^n dt
所以:
∫(上限1,下限0) x^n *(1-x)^m dx
= 令t=x-(1/2), ∫(上限1/2,下限-1/2) ((1/2)+t)^n *((1/2)-t)^m dt
= -∫(上限-1/2,下限1/2) ((1/2)-t)^n *((1/2)+t)^m dt
= ∫(上限1/2,下限-1/2) ((1/2)+t)^m *((1/2)-t)^n dt
=∫(上限1,下限0) x^m *(1-x)^n dx
= 令t=x-(1/2), ∫(上限1/2,下限-1/2) ((1/2)+t)^m *((1/2)-t)^n dt
所以:
∫(上限1,下限0) x^n *(1-x)^m dx
= 令t=x-(1/2), ∫(上限1/2,下限-1/2) ((1/2)+t)^n *((1/2)-t)^m dt
= -∫(上限-1/2,下限1/2) ((1/2)-t)^n *((1/2)+t)^m dt
= ∫(上限1/2,下限-1/2) ((1/2)+t)^m *((1/2)-t)^n dt
=∫(上限1,下限0) x^m *(1-x)^n dx
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