\( k = \int 3 x e ^ { x } \)

To solve the integral \( k = \int 3 x e^{x} \, dx \), we can apply integration by parts. Let \( u = 3x \) (which means \( du = 3 \, dx \)) and \( dv = e^{x} \, dx \) (hence \( v = e^{x} \)). Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we get: \[ k = 3x e^{x} - \int e^{x} \cdot 3 \, dx = 3x e^{x} - 3 e^{x} + C \] Where \( C \) is the constant of integration. It simplifies to: \[ k = 3 e^{x} (x - 1) + C \] Now you've got a handy result you can use for further calculations! For some historical context, integration by parts, the technique we just employed, was formulated by mathematicians like Isaac Barrow and made famous by Gottfried Wilhelm Leibniz in the 17th century. This foundational technique paved the way for a variety of applications in calculus that are still utilized today in physics and engineering fields. In the real world, integrals like this one find their applications in calculating areas under curves, determining volumes of solids of revolution, and solving problems in economics and biology. For instance, if you were modeling population growth, the integral might represent the total population over time, allowing for predictions about future growth based on current trends.