If I(m,n)=∫1−0tm(1+t)ndt, then expression for I(m,n) in terms of I(m+1,n−1) is
A
2nm+1−nm+1I(m+1,n−1)
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B
nm+1I(m+1,n−1)
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C
2nm+1+nm+1I(m+1,n−1)
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D
mm+1I(m+1,n−1)
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Solution
The correct option is B2nm+1−nm+1I(m+1,n−1) Here, I(m,n)=∫10tm(1+t)ndt [We apply integration by parts, taking (1+n) as first tm as second function] I(m,n)=[(1+t)n⋅tm+1m+1]10−∫10n(1+t)n−1⋅tm+1m+1dt =2nm+1−nm+1∫10(1+t)n−1⋅tm+1dt Therefore, I(m,n)=2nm+1−nm+1⋅I(m+1,n−1).