1

Let $f$ be a $C^1$ function with $f'>0$, and let $a\not= 0$ be a real number. Is there a closed form for the integral $$ \int f(x) f'(x)^a \mathrm dx? $$

Certainly if $a=1$, then the integral is simply $f^2/2 + c$, but I do not see a way of doing it for arbitrary $a$

MSDG
  • 7,143
  • As a note, the “good twin” $f^af’$ has an easy antiderivative, but I guess your question originates from that – b00n heT Sep 02 '19 at 13:36
  • @b00nheT Thanks for the comment. Actually these integrals appeared somewhat randomly in a project I am working on, and it would be nice to have a closed form for them, but I guess this might not be possible =) – MSDG Sep 02 '19 at 13:41

1 Answers1

1

No, there is no closed form for that. Indeed, try this one $f(x) = x\log x$, $a=1/2$: $$ \int x \log x \sqrt{1+\log x}\;dx $$

New example with $f$ defined on all of $\mathbb R$ and $f'>0$. Take $f(x) = x+e^x, a=1/2$. $$ \int(x+e^x)\sqrt{1+e^x}\;dx $$

GEdgar
  • 111,679
  • Nice example, thanks. I am also assuming that $f$ is defined on all of $\mathbb{R}$ and satisfies $f'>0$ for all reals (I was a bit unclear with this in the post), but I suppose that there might not be a closed form even assuming that. – MSDG Sep 02 '19 at 13:34