Evaluate the following integral using integration by parts. \[ \int e^{-14 s} \mathrm{ds} \]

To evaluate the integral \( \int e^{-14s} \, ds \) using integration, we actually don't need to use integration by parts since this integral can be directly computed. However, since the inquiry suggests seeking a methodical approach, let’s explore how we would apply the basic principles of integration. 1. We start with the integral: \[ \int e^{-14s} \, ds \] 2. The integral of \( e^{kx} \) is \( \frac{1}{k} e^{kx} + C \), where \( C \) is the constant of integration. Here, \( k = -14 \). 3. Applying this rule, we have: \[ \int e^{-14s} \, ds = \frac{1}{-14} e^{-14s} + C \] 4. Thus, the final answer becomes: \[ -\frac{1}{14} e^{-14s} + C \] And there you have it! The integral evaluates to \( -\frac{1}{14} e^{-14s} + C \).