While I was playing with Wolfram Alpha online calculator I wondered how to calculate and justify an approximation of an integral involving the fractional part function, the exponential and the gamma function $$\int_0^1\left\{\frac{e^x}{\Gamma(x)}\right\}dx. $$ Thus here $\left\{x\right\}$ denotes the fractional part function, see these code
int frac(e^(x)/Gamma(x))dx, from x=0 to 1
and
plot e^x/Gamma(x), from x=0 to 1
I believe that it is feasible get the good approxiation that I've evoked using (real) analysis, but I am not sure of my strategy.
Question. What is a good method to calculate and justify a good approximation of the integral $$\int_0^1\left\{\frac{e^x}{\Gamma(x)}\right\}dx\,?$$ Many thanks.