Put and your integral becomes
See some of
Gamma function - Wikipedia, the free encyclopedia
im having a lot of trouble showing that the
definite integral from 0 to 1 (ln x)^n dx = n!(-1)^n
if someone could help get me started cause i have no idea how to get the factorial in the answer ... all i get is this infinite loop of integration by parts which keeps going on and on
i am not familiar with gamma functions ... is there an easier way of solving this problem ... maybe this will help
there was a previous part to this question which i was able to solve
I(subscript)n = integral (ln x)^n dx implies that
I(subscript)n = x (ln x)^n - n*(I(subscript)n-1)
maybe i have to somehow use this to solve the second question ... but i cannot see anyway to use it ...
The way Kriz showed is about the easiest. If you don't know the Gamma, then get to know it. It's a great tool to have in the arsenal.
Here's another subtle difference in the sub Kriz mentioned.
Make the subs and change the limits of integration.
When x=1, then t=0. When x=0, then
Factor out the and reverse the limits of integration to shed the negative sign.
Then we have for ours, which equals n!
So, we get