Is $ \int \frac{f'(x)}{f(x)} \ dx = \log |f(x)| + C$ true for all differentiable functions $f$?

Question

Let $f$ be a differentiable function. Is the following identity true for all such $f$? $$ \int \frac{f'(x)}{f(x)} \ dx = \log |f(x)| + C $$

I ask because there exist differentiable functions whose derivatives are not Riemann integrable (see here for instance). On the other hand, if we use the substitution $u = f(x)$ for $f$ on $[a,b]$, $$ \int_a^b \frac{f'(x)}{f(x)} \ dx = \int_{f(a)}^{f(b)} \frac{1}{u} \ du $$ and the RHS appears to be integrable. How can we reconcile this?

Any comments, help and explanations are welcome.

Answer 1

If you wonder what goes wrong when doing the substitution, it is because the conditions needed for doing an integration by substitution are not met. Indeed (using your notation), if $f\colon [a,b]\rightarrow I$ is a differentiable function with a continuous derivative, and $g\colon I\rightarrow \mathbb{R}$ is continuous (and $I$ is an interval), then $$\int_a^b g(f(x))f'(x) dx = \int_{f(a)}^{f(b)} g(u) du.$$ In your case $f$ does not have a continuous derivative, so we cannot expect this equality to hold.

Answer 2

Even if you take, say $f(x)=2+x^2\sin\frac{1}{x^2}, f(0)=2$ on $[-1,1]$ so that $f(x)\neq 0,$ you have $f’(x)$ is unbounded and $f$ is bounded away from $0$ on $[-1,1],$ so $\frac{f’(x)}{f(x)}$ is not bounded, and hence it isn’t Riemann integrable.

Indeed, if $f$ is never $0$ on $[a,b]$ then $f$ is bounded away from zero, so that $f’(x)$ is unbounded if and only if $f’(x)/f(x)$ is unbounded.

That doesn’t mean there isn’t a useful value for the integral, only that Riemann doesn’t get us to there immediately.

For the given $f,$ we’d have to do $$\int_{-1}^{-a}+\int_b^1$$ and then let $a,b\to0^+,$ to get a the value.

Or we can do a Lesbesgue integral.

Either way, we’ll get $\log |f(1)|-\log|f(-1)|.$

Answer 3

Be careful too. If $f(x) = 0$ for any $x \in [a,b]$, the integral won't be defined.