We will do the integral of sin(ln(x))/ln(x) from 0 to 1 by using Feynman's Trick (aka differentiation under the integral sign). This is also closely related to the Dirichlet Integral, which is the integral of sin(x)/x from 0 to inf.
For the complex exponential definition of sine    • Complex definitions of sine and cosine  
For another integral by Feynman's Technique:    • Feynman technique: integral of (x1)/...  

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