\( \int_{1}^{64}\left(\frac{1 \cos ^{2}(x)}{1}\left(\right.\right. \) fracl \( \left.\cos ^{2}(x)\right)( \) fracsin \( (x) \cos (x)) d \)

It seems like there are some formatting issues in your integral expression. However, I will try to interpret it and provide a solution based on what I can understand. If you meant to evaluate the integral of the form: \[ \int_{1}^{64} \frac{\cos^2(x)}{\sin(x) \cos(x)} \, dx \] we can simplify the integrand. The expression \(\frac{\cos^2(x)}{\sin(x) \cos(x)}\) can be rewritten as: \[ \frac{\cos^2(x)}{\sin(x) \cos(x)} = \frac{\cos(x)}{\sin(x)} = \cot(x) \] Thus, the integral becomes: \[ \int_{1}^{64} \cot(x) \, dx \] The integral of \(\cot(x)\) is: \[ \int \cot(x) \, dx = \ln|\sin(x)| + C \] Now, we can evaluate the definite integral: \[ \int_{1}^{64} \cot(x) \, dx = \left[ \ln|\sin(x)| \right]_{1}^{64} \] Calculating this gives: \[ \ln|\sin(64)| - \ln|\sin(1)| \] This can be simplified to: \[ \ln\left(\frac{|\sin(64)|}{|\sin(1)|}\right) \] To find the numerical values of \(\sin(64)\) and \(\sin(1)\), you can use a calculator or software that computes trigonometric functions. If you meant a different integral or if there are additional details, please clarify, and I would be happy to assist further!