At a Glance - In the Real World
We'll be honest: a lot of the mechanical integration methods you're learning here probably won't be that useful in the long run. Once you get out of school and into a real-life situation, you'll get to use computers most of the time.
The most useful aspect of the integration problems isn't the integration. It's the practice you get at figuring out how to attack each new problem and which integration technique to use. This will make you generally better at figuring out what to do when you encounter new types of problems.
The improper integrals are more useful. Improper integrals play a large role in the study of probability, once we get beyond problems like "what is the likelihood of picking a blue sock out of the drawer?" Improper integrals are also used to create the Fourier transform and the Laplace transform, which physicists and engineers use to help solve certain types of differential equations.