Numerical integration is a method used to approximate the value of an integral when it cannot be solved analytically. In this case, we are trying to solve the integral from 10 to positive infinity of (sin(1/x)/(1+x^3))dx with an error precision of e=0.5*10^-4. This means that we want our approximation to be within 0.5*10^-4 of the actual value of the integral.
To solve this using Simpson's rule, we need to first divide the interval of integration (10 to infinity) into smaller subintervals. Simpson's rule works by approximating the curve of the function with a series of parabolas, so we need to have an even number of subintervals. Let's say we divide the interval into n subintervals, each with a width of h.
Next, we need to calculate the values of the function at the endpoints of each subinterval. In this case, our function is (sin(1/x)/(1+x^3)), so we need to calculate f(10), f(10+h), f(10+2h), and so on until we reach the endpoint of the last subinterval. We can use a calculator or a computer program to do this.
Once we have all the function values, we can use Simpson's rule formula to approximate the value of the integral. The formula is:
I ≈ (h/3) * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + 4f(xn-1) + f(xn)]
where x0 is the starting point of the interval (10 in this case), xn is the endpoint of the interval (infinity in this case), and h is the width of each subinterval (h = (infinity - 10)/n).
To meet our error precision of e=0.5*10^-4, we need to choose a value of n that will make the error term (h^4/180)*f^(4)(ξ) (where ξ is some value between 10 and infinity) less than or equal to 0.5*10^-4. This will require some trial and error, but we can start by choosing a value of n and calculating the error term. If it is larger than 0.5*
