How can I express $\int \frac{1}{f'(x)}$ in terms of $f(x)$

Question

More specifically, I would like to know if there is a way I can express

$$\int \frac{x g'(x)}{f'(x)} dx $$

In terms of $f(x)$ and $g(x)$. Both $f(x)$ and $g(x)$ are non-negative and known to be increasing in $x$. If that helps.

Answer 1

You cannot expect to be able to achieve a closed form solution for every choice of $f(x)$ and $g(x)$.

For example, take $f(x)=\frac{x^3}{3}$ and $g(x)=e^x$ so that $f'(x)=x^2$ and $g'(x)=e^x$.

Now:

$$\displaystyle\int\frac{xg'(x)}{f'(x)}dx=\int\frac{e^x}{x}dx.$$

This is known to have no solution in terms of elementary functions: see the Wikipedia entry on the exponential integral

Edit:

You can also use a similar construction to try and show that you can't expect to be able to find $\int \frac{1}{f'(x)}dx$ simply. Set $f(x)=-\frac{x+1}{e^x}$ so that $f'(x)=\frac{x}{e^x}$, and you arrive at the same integral as above.