1. **Problem:** Evaluate the integral $$\int \left(\frac{\sec x}{\csc x} + \frac{\csc x}{\sec x}\right) dx$$. 2. **Formula and rules:** Recall that $$\sec x = \frac{1}{\cos x}$$ and $$\csc x = \frac{1}{\sin x}$$. 3. **Rewrite the integrand:** $$\frac{\sec x}{\csc x} + \frac{\csc x}{\sec x} = \frac{\frac{1}{\cos x}}{\frac{1}{\sin x}} + \frac{\frac{1}{\sin x}}{\frac{1}{\cos x}} = \frac{1}{\cos x} \cdot \frac{\sin x}{1} + \frac{1}{\sin x} \cdot \frac{\cos x}{1} = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}$$ 4. **Simplify:** $$\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \tan x + \cot x$$ 5. **Integral becomes:** $$\int (\tan x + \cot x) dx = \int \tan x \, dx + \int \cot x \, dx$$ 6. **Recall integrals:** $$\int \tan x \, dx = -\ln|\cos x| + C$$ $$\int \cot x \, dx = \ln|\sin x| + C$$ 7. **Combine results:** $$\int (\tan x + \cot x) dx = -\ln|\cos x| + \ln|\sin x| + C = \ln\left|\frac{\sin x}{\cos x}\right| + C = \ln|\tan x| + C$$ **Final answer:** $$\int \left(\frac{\sec x}{\csc x} + \frac{\csc x}{\sec x}\right) dx = \ln|\tan x| + C$$