Extract function from integrating by parts

Question

It is known that:

$\int u(t) v'(t) \,dt=u(t)v(t)-\int u'(t)v(t)\,dt$

Is it possible to "bracket out" the function $v(t)$ on the right side, i.e .:

$u(t)v(t)-\int u'(t)v(t)\,dt=(???) \cdot v(t)$

Answer 1

Assume this was possible and consider some non-elementary integrable function $f$. Let $u(x) = x, v = f$ and denote your $(\dots)$ by $h(x)$. Furthermore let's assume that $h$ is an elementary function because otherwise it is of course trivially possible but of no real use. Then $$ u(x)v(x) - \int u'(x) v(x) dx = x f(x) - \int f(x) dx = h(x) \cdot f(x) \implies \int f(x) dx = (x - h(x)) f(x) $$ which would be an elementary function for the integral of $f$. But this contradicts our assumption. Thus it is in general not possible to do this.