   Chapter 7.P, Problem 8P

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# If n is a positive integer, prove that ∫ 0 1 ( ln x ) n d x = ( − 1 ) n n !

To determine

To prove:

That if n is a positive integer then 01(lnx)ndx = n!(1)n.

Explanation

Given:

01(lnx)ndx

Formulae used:

Integration by parts

udv=uv-vdu

Consider 01(lnx)ndx

Now, use integration by parts with u=(ln x)n and dv=dx. Then du=n(ln x)n-11/x dx and v=x. Thus we have

01(lnx)ndx=[(lnx)nx]0101xd(lnx)n=001xn(lnx)n11xdx=(n)01(lnx)n1dx

Continuing similarly we shall have,

(n)01(lnx)n1dx=(n)((n1))01(lnx)n2dx=(n)((n1))((n2))01(lnx)n3dx==(1)nn(n1)(n2)

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