Question

What is the value of \(\int_0^1 log(y)^8 dy\)?

(a) 5!

(b) 6!

(c) 7!

(d) 8!

I have been asked this question in examination.

Question is from Special Functions topic in chapter Special Functions – Gamma, Beta, Bessel and Legendre of Engineering Mathematics

Answer 1

Right answer is (d) 8!

The best I can explain: \( \Gamma(n) = \int_0^1 log \left(1 \atop y\right)^{n-1} \)

Here, the integral is \( \int_0^1 log (y)^8 dy\) which can also be written as \( \int_0^1- log(y)^8 dy \) which is actually \(\int_0^1 log \left(1 \atop y\right)^{9-1} = \Gamma(9) = 8!.\)