1. **Problem statement:** Evaluate the integral $$\int \frac{5}{4 - \sqrt{3} - z} \, dz$$. 2. **Formula and approach:** This is a rational function integral of the form $$\int \frac{A}{B - z} \, dz$$ where $A$ and $B$ are constants. 3. **Step-by-step solution:** - Rewrite the integral as $$5 \int \frac{1}{4 - \sqrt{3} - z} \, dz$$. - Let $$u = 4 - \sqrt{3} - z$$, then $$du = -dz$$ or $$dz = -du$$. - Substitute into the integral: $$5 \int \frac{1}{u} (-du) = -5 \int \frac{1}{u} \, du$$. - The integral of $$\frac{1}{u}$$ is $$\ln|u|$$, so: $$-5 \ln|u| + C = -5 \ln|4 - \sqrt{3} - z| + C$$. 4. **Final answer:** $$\boxed{-5 \ln|4 - \sqrt{3} - z| + C}$$ --- **Note:** The other integrals were not solved as per instructions to solve only the first problem.