Here we will introduce several theoretical results that lie behind most methods for evaluating indefinite integrals. However, there is a big difference when compared to the situation with limits and derivatives. There we had theorems that specified how the relevant notion (limit, derivative) deals with typical situations (algebraic operations, composition). Just knowing these theorems we already obtained a practical approach to calculations that was pretty much sufficient for typical problems, so in the end it was essentially an algorithmic affair.
There are no such theorems for integrals. We do have linearity, but there are no rules for a product, division nor a composed function, hence there is also no algorithm for integration. We find indefinite integrals using various tricks, which will be covered in Methods Survey - Integration and which often use the results that we will present below. Thus a good knowledge of these sections is essential for being able to integrate, for practice we recommend problems marked as "simple" in Exercises.