Proof of a closed form of \(\displaystyle\int^{{1}}_{0}{\left(-{\ln{{x}}}\right)}^{{n}}{\left.{d}{x}\right.}\)

Petrolovujhm

Petrolovujhm

Answered question

2022-04-02

Proof of a closed form of 01(lnx)ndx

Answer & Explanation

kaosimqu5t

kaosimqu5t

Beginner2022-04-03Added 10 answers

Take the integral
I(a)=01xadx=1a+1
Now take the derivative n times and obtain
I(a)(n)=(1)!(a+1)n+1,
which gives
I(0)(n)=(1)!.
On the other hand differentiating under the integral sign gives
I(0)(n)=01ln(x)ndx
And so conclude the result.

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