Question

another defination of n surprise_(factorial)

Answer 1

prove \[\int _0^1 (\ln x)^ndx=n!\]

Answer 2

surprise!!!!!

Answer 3

\[\int_0^1(\ln x)^n dx= n!!!!!!!!!!!!!!\]

Answer 4

\[\Large u=\ln x, e^u=x, e^udu=dx\]
\[\large\int_0^1(\ln x)^ndx=\int_0^1u^ne^udu=\fbox{stuck xD}\]

Answer 5

is the fact that ln x is not continuous on x=0 a problem

Answer 6

my guess is induction plus parts, but i could be wrong

Answer 7

for this questionit says n is \[\Large even\]

Answer 8

ok well that is good because
\[\int_0^1ln(x)dx=-1\]

Answer 9

by parts\[f=\ln^n x,g=1\\\int x\ln^nxdx=x \ln^n x|_0^1-n\int_0^1 x(\ln^{n-1}x)\frac{1}{x} dx \]
\[I_n=-nI_{n-1}\implies I_n=-n(-(n-1)I_{n-2})=...\]

Answer 10

\[\Large\int_0^1(\ln x)^ndx
\\\Large=\left[x(\ln x)^n-\int xd(\ln x)^n\right]_0^1
\\\Large=\left[x(\ln x)^n-\int n(\ln x)^{n-1}dx\right]^1_0
\\\Large=\left[x(\ln x)^n-nx(\ln x)^{n-1}+\int n(n-1)(\ln x)^{n-2}dx\right]_0^1
\\\Large=\cdots
\\\Large
\]

Answer 11

\[I_n=(-1)^nn(n-1)(n-2)...3*2*1=n!,\\if\\n=2k\]