Problem $2$: Thermal Conductivity in a Material $($Engineering$)$

Background: In engineering, understanding the thermal conductivity of materials is crucial for heat transfer analysis. The heat transfer rate can vary with temperature and distance, especially in materials where thermal conductivity changes with the environment. Here, we consider a scenario where the heat transfer rate is modeled with an exponential function, which requires integration by parts for calculation.

Formula: The total heat transfer $Q$ over a distance is given by:

$Q={\displaystyle \underset{0}{\overset{L}{}}}k(x)e^{-x}dx$

where $k(x)$ is a function describing the change in thermal conductivity along the length of the material.

Problem: Calculate the total heat transfer over a material of length $L=3,$ where the thermal conductivity function is given by $k(x)=x^2.$

$[12$pt$]$ A$.$ Evaluate the integral using at least two different integration techniques.

$[3$pt$]$ B$.$ Reflect on the two methods. Which one provided a clearer path to the solution, and why?