Question

Are there functions that cannot be integrated using integration by parts?

Answer 1

Yes, there are infinitely many functions that cannot be integrated with a close form integral.

Explanation:

If you can integrate `int f(x) dx` at all, then you can, in a trivial sense, integrate by parts.

`u = 1` and `dv = f(x) dx` so `du = 0 dx` and `v = int f(x) dx`

`uv-intvdu = 1intf(x) dx - int [int f(x) dx] 0 dx`

Examples of integrals without closed form include

`inte^(x^2) dx`, `int cosx/x dx` and `int sinx/x dx`