Can substitutions in indefinite integral make it non elementary?

Question

I was thinking that if we make a substitution in an elementary indefinite integral ,can this substitution convert that integral into a non elementary/non evaluatable one ?. Till date i have never faced such a substitution .Can someone give an example for it ?

Answer 1

Briefly, it's impossible: If $F' = f$, then substituting $u = g(x)$ and $du = g'(x)\, dx$ effects the same substitution on the antiderivatives. That is, omitting the constants of integration, $$ F(g(x)) = \int f(g(x))g'(x)\, dx = \int f(u)\, du = F(u). $$