How do you determine the indefinite integrals?
There is no one method for finding indefinite integrals.
There are many methods used, but no one method for all integrals. A problem the can be solved by integration by parts (reversing the product rule) may or may not be solvable by integration by u-substitution (reversing the chain rule). Other problems require other techniques.
Some functions have indefinite integrals that cannot be finitely expressed without the integration symbol.
Introductory calculus courses typically include several techniques of integration in their syllabus.