<s>[INST] How to evaluate $\int_0^1 \frac{1-x}{\ln x}(x+x^2+x^{2^2}+x^{2^3}+x^{2^4}+\ldots) \, dx$? Evaluate the definite integral:
$$\int_0^1 \frac{1-x}{\ln x}(x+x^2+x^{2^2}+x^{2^3}+x^{2^4}+\ldots) \, dx$$
I think the series involving $x$ converges because $x\in[0,1]$, but I cannot form an expression for the series. If I let
$$
u_n=x^{2^{n-1}} \\ \frac{\ln u_n}{\ln x}=2^{n-1}
$$
but then this series does not converge. Even WolframAlpha cannot evaluate a definite integral together with an infinite series, so I am stuck on this.
 [/INST] Claim 1: For $k\geq 1$, we have that
\begin{align}
\int^1_0 \frac{1-x}{\log x}x^{2^k}\ dx = -\log \frac{2^k+2}{2^k+1}.
\end{align}
Claim 2: We have
\begin{align}
\prod^\infty_{k=0}\left( 1+\frac{1}{2^k+1}\right) = 3
\end{align}
Using the claims, we have the series
\begin{align}
-\sum^\infty_{k=0} \log \left(\frac{2^k+2}{2^k+1}\right) =-\log\left(\prod^\infty_{k=0} \left(1+\frac{1}{2^k+1}\right) \right) =-\log 3.
\end{align}
</s>