1. ## Reduction formulae

i) Given that $In= \int_{1}^{0} (lnx)^n dx$, where n is an integer, prove that

$In = -nI(n-1)$

ii) Hence find In in terms of n.

I can do the first part.
How do i do the second part?

thank you.

2. ## Re: Reduction formulae

Note that $I_0=-1$, write several first values of $I_n$ and try to guess the general formula for $I_n$.

3. ## Re: Reduction formulae

In= -n*-(n-1)*-(n-2)....
is this correct?

4. ## Re: Reduction formulae

$I_n=(-1)^{n+1}n!$

5. ## Re: Reduction formulae

can you explain step by step?
im feeling stupid

thank you.

6. ## Re: Reduction formulae

Starting from $I_0=-1$ and calculating subsequent values of $I_n$ using the recurrence equation $I_n=-nI_{n-1}$, we get

$\begin{array}{c|c|c|c|c|c|c}n & 0 & 1 & 2 & 3 & 4 & 5\\\hline I_n & -1 & 1 & -2 & 6 & -24 & 120\end{array}$

As you noted, $|I_n|=n(n-1)(n-2)\cdot\ldots\cdot1$, which, by definition, is $n!$. Further, we note that $I_n>0$ for odd n and $I_n<0$ for even n. The sequence $(-1)^{n+1}$ shows a similar behavior with respect to alternating signs. Altogether, $I_n=(-1)^{n+1}n!$.