Need help with numerical integration

Im supposed to solve
integral 10 to +infinity ((sin(1/x)/(1+x^3))dx with error precision of e=0.5*10^-4. Can someone please give me detailed explenation of solving this. (Supposedly by Simpson but i get lost in the way.

P.S. sorry for bad spelling and lack of proper formula notions.
I process from integral i get
1/3[ln((x+1)/sqrt(x^2-x+1))+sqrt(3)*arctg(2*sqrt(3)*x/3-sqrt(3)/3)] from M to +infinity <=1/4*10^-2

than i get

-ln((M+1)/(sqrt(M^2-M+1))) +sqrt(3)*(pi/2-arctg(2*sqrt(3)*M/3-sqrt(3)/3))<=7.5*10^-3

and i cant find any exact solution to solve that

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