Evaluate $\displaystyle \int_0^1\frac{\ln(x+1)}x\,dx$.
How many methods you know to solve it?
$\displaystyle \ln (x+1) = \sum_{n=1}^{\infty} \frac{(-1)^nx^n}{n}$.
Thus, $\displaystyle \frac{\ln (x+1)}{x} = \sum_{n=1}^{\infty} \frac{(-1)^n x^{n-1}}{n}$.
This means, $\displaystyle \int_0^1 \left( \sum_{n=1}^{\infty} \frac{(-1)^n x^{n-1}}{n}\right) dx = \sum_{n=1}^{\infty} \left( \int_0^1 \frac{(-1)^{n+1} x^{n-1}}{n} dx \right) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}= \frac{\pi^2}{12}$
This particular integral is related to what is known as the 'dilogarithm'.
$\displaystyle dilog(x)=\int_{1}^{x}\frac{ln(t)}{t-1}dt$
In our case, $\displaystyle \int_{0}^{1}\frac{ln(x+1)}{x}dx=\int_{1}^{2}\frac{ ln(x)}{x-1}dx = -dilog(x+1) = \frac{{\pi}^{2}}{12}$
It seems to me there is also a unit disk reference with the dilog. It says for n=2 and z in the unit disk, then
$\displaystyle Li_{2}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{2}}$
i.e, if z=1, then we have $\displaystyle \frac{{\pi}^{2}}{6}$
There is also a trilogarithm.
Do a google to find out more. It is a cool function with lotsa implications.
For instance, it has the series: $\displaystyle -x-\frac{x^{2}}{4}-\frac{x^{3}}{9}-.....-\frac{x^{n}}{n^{2}}$
Also, $\displaystyle Li_{2}(z)=\int_{z}^{0}\frac{ln(1-t)}{t}dt$
In the event z = -1, then we have what PH posted.
Notice how the Basel sum is involved here. $\displaystyle \sum_{k=1}^{\infty}\frac{1}{n^{2}}=\frac{{\pi}^{2} }{6}$
I noticed the similarity between this and the posters integral and thought there was a connection. I am sure Wiki or Mathworld has something about it.
The dilog is the polylog with n=2. There are others. As I mentioned, a trilog.
It has implications to gamma and other things. Research it. You'll find it cool.
$\displaystyle \sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n + 1} }}
{{n^2 }}} = 1 - \frac{1}
{{2^2 }} + \frac{1}
{{3^2 }} - \frac{1}
{{4^2 }} + \cdots .$
This last is equal to
$\displaystyle \left( {1 + \frac{1}
{{2^2 }} + \frac{1}
{{3^2 }} + \frac{1}
{{4^2 }} + \cdots } \right) - 2\left( {\frac{1}
{{2^2 }} + \frac{1}
{{4^2 }} + \cdots } \right).$
Since $\displaystyle 2\left( {\frac{1}
{{2^2 }} + \frac{1}
{{4^2 }} + \cdots } \right) = \frac{2}
{{2^2 }}\left( {1 + \frac{1}
{{2^2 }} + \cdots } \right),$
we have $\displaystyle \sum\limits_{n = 1}^\infty {\frac{{( - 1)^{n + 1} }}
{{n^2 }}} = \sum\limits_{n = 1}^\infty {\frac{1}
{{n^2 }}} - \frac{1}
{2}\sum\limits_{n = 1}^\infty {\frac{1}
{{n^2 }}} = \frac{{\pi ^2 }}
{6} - \frac{{\pi ^2 }}
{{12}} = \frac{{\pi ^2 }}
{{12}}.$