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f(n)=int(0)^(oo)x^(n-1)e^(-x)dx...

`f(n)=int_(0)^(oo)x^(n-1)e^(-x)dx`

A

`(f(n+1))/((m+1)^(n))`

B

`(f(n))/((m+1)^(n+1))`

C

`(f(n+1))/((m+1)^(n+1))`

D

`g(m+1),n+1)`

Text Solution

Verified by Experts

The correct Answer is:
C
Putting `log_(e)..(1)/(x)=t`
`rArr" "x=e^(-t)`
`rArr" "int_(0)^(1)x^(m)(log_(e)(1)/(x))^(n)dx`
`" "=int_(oo)^(0)e^(-mt)t^(n)(-e^(t))dt`
`" "=int_(0)^(oo)t^(n)e^(-(m+1))dt`
`" "=(1)/((m+1)^(n+1))int_(0)^(oo)t^(n)e^(-y)dy" (putting (m + 1) t = y)"`
`" "=(f(n+1))/((m+1)^(n+1))`
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