Exactly. In fact, substitution is the chain rule in reverse. The reason we have to do substitution is to make the integral you are working on

**exactly** match a known integral, not just look generally like it.

Incidentally, many tables of integrals give a more general form than the one you presumably know. For example, see #16 and #17

here. What they have done here is to "pre-digest" the substitution for you in #17. (Both can be done by trig substitution, which is no different from substitution in general except that you are replacing a variable with an expression rather than an expression with a variable, and you have to learn to recognize which one will work.)