# integration – Integral from Mathematica’s documentation: \$int_0^1 frac{log left(frac{1}{2} left(1+sqrt{4 x+1}right)right)}{x} , dx = frac{pi^2}{15} \$ I like to puruse Mathematica’s documentation and look at the ‘Neat Examples’: this is one I managed to figure out. Apparently it’s due to Ramanujan:
$$I=int_0^1 frac{log left(frac{1}{2} left(1+sqrt{4 x+1}right)right)}{x} , dx = frac{pi^2}{15}$$

Here are the steps for my solution:

1. Make the substitution $$x=y^2-y$$, yielding
$$I= int _{1}^{phi}frac{log(y)(2y-1)}{y(y-1)},dy,$$where $$displaystyle{phi = frac{1+sqrt{5}}{2}}$$ is the golden ratio.
2. Factor out the $$log(y)$$ term and use partial fractions to write
$$I = underbrace{int _{1}^{phi}frac{log(y)}{y},dy}_{I_1} + underbrace{int _{1}^{phi}frac{log(y)}{y-1},dy}_{I_2}$$$$I_1$$ can be evaluated using a simple substitution, yielding $$displaystyle{I_1 = frac{log ^2(phi )}{2}}$$.
3. Use the Taylor series for $$log(y)$$ centered at $$y=1$$ and interchange the sum and integral to show
$$I_2 = -sum_{k=1}^{infty} frac{(1-phi)^{k}}{k^2}= -sum_{k=1}^{infty} frac{(-phi^{-1})^{k}}{k^2}= – text{Li}_2(-phi^{-1})$$
4. $$text{Li}_2$$ has the following properties:
• $$text{Li}_2(x) + text{Li}_2(-x) = frac{1}{2}text{Li}_2(x^2)$$
• $$text{Li}_2(x) + text{Li}_2(1-x) = zeta(2) – log(x)log(1-x)$$
• $$text{Li}_2(1-x) + text{Li}_2(1-x^{-1}) = -frac{1}{2}log^2(x)$$

Put $$x=phi^{-1}$$ and use $$phi^2=phi+1$$; this gives:
$$text{Li}_2(phi^{-1}) + text{Li}_2(-phi^{-1}) = frac{1}{2}text{Li}_2(1-phi^{-1})$$
$$text{Li}_2(phi^{-1}) + text{Li}_2(1-phi^{-1}) = zeta(2) -2 log^2(phi)$$
$$text{Li}_2(1-phi^{-1}) + text{Li}_2(-phi^{-1}) =-frac{1}{2}log^2(phi)$$
5. Relabel for clarity. Let $$A=text{Li}_2(phi^{-1})$$, $$B=text{Li}_2(-phi^{-1})$$, $$C=text{Li}_2(1-phi^{-1})$$, and $$L= log^2(phi)$$. This gives the system
$$begin{cases} A+ B & = frac{1}{2}C\ A+ C&= zeta(2)- 2L\ C+B &= -frac{1}{2}L end{cases}$$Solving gives $$B=-I_2=displaystyle{frac{1}{2}L-frac{2}{5}Z}$$, whence $$displaystyle{I = frac{pi^2}{15}}.$$

I’d be curious to see if there are any other methods of proof, perhaps involving simpler substitutions than the ones I used. Posted on Categories Articles