such proofs usually use ideas well beyond the scope of calculus texts. for example:
one common reference given for such proofs (and indeed, quoted in the link above) is: M. Rosenlicht, Integration in finite terms, American Math. Monthly 79 (1972), 963-972.
don't feel daunted if either of these is beyond what you have learned so far. it turns out that "integration in elementary terms" is not an elementary subject.