MHB Solve Exponential Integral: \int \frac{2^{x}\cdot 3^{x}}{9^{x}-4^{x}}dx

Yankel
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Hello

I am trying to solve this exponential integral, it's quite complicated. Any hints ?

\int \frac{2^{x}\cdot 3^{x}}{9^{x}-4^{x}}dxmany thanks
 
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Start by the following

\int \frac{ 1}{\frac{3^{x}}{2^x}-\frac{2^{x}}{3^x}}dx
 
For original Zeta function, ζ(s)=1+1/2^s+1/3^s+1/4^s+... =1+e^(-slog2)+e^(-slog3)+e^(-slog4)+... , Re(s)>1 Riemann extended the Zeta function to the region where s≠1 using analytical extension. New Zeta function is in the form of contour integration, which appears simple but is actually more inconvenient to analyze than the original Zeta function. The original Zeta function already contains all the information about the distribution of prime numbers. So we only handle with original Zeta...
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