Evaluate the following integral. \[ \int_{\frac{1}{3}}^{\frac{1}{2}} \frac{11^{\frac{1}{p}}}{p^{2}} d p \] \[ \int_{\frac{1}{3}}^{\frac{1}{2}} \frac{11^{\frac{1}{p}}}{p^{2}} d p=\square \text { (Type an exact answer.) } \]

Step 1 ：This is a definite integral problem. The integral is from 1/3 to 1/2 of the function \(\frac{11^{\frac{1}{p}}}{p^{2}}\).

Step 2 ：To solve this, we can use numerical integration methods such as the trapezoidal rule, Simpson's rule, or Romberg's integration.

Step 3 ：Using these methods, we find that the integral is approximately 504.6091936233386.

Step 4 ：Final Answer: The value of the integral is approximately \(\boxed{504.6091936233386}\).