We reviewed for a test today. It focuses on anti-derivatives (indefinite integrals).
There are a few options for approaching the problems. With u-substitution, you should be able to see fairly soon if it’s the right approach. You can also go term by term. Many times, rewriting and/or doing some algebra is helpful beforehand.
So when you’re first doing integrals, you have a limited toolbox. You will expand that as you go and soon see that integration can be much more complicated than taking a derivative. But at first, you just use a few tools.
There used to be and probably still are mathematicians that spend years solving unsolved integrals. There are books filled with tables of integrals.