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Is there any rule for integrals of the form $$\int f(g′(x))g(x)\,dx.$$

Motivated by Where is the form $f(g(x))g'(x)$? and Integration of f(g(x)) by substitution, "creating" g'(x) dx term, how it works?

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    No. There's no reason why there should be. The one you cite is true because of the chain rule. There's no corresponding fact about differentiation to justify the one you hope for. – Ethan Bolker May 08 '17 at 15:24
  • Maybe logarithmic derivative can give a hint: $\frac {d\log(f(x))}{d x} = \frac{f'(x)}{f(x)}$. – mathreadler May 08 '17 at 15:34

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The reason why integrals of the form $$\int f\left(g(x)\right) g'(x) \, \mbox{d}x$$ can be simplified to $$\int f\left(u\right) \, \mbox{d}u$$ is a direct consequence of the chain rule for derivatives.

There is no simple rule (for derivatives) which would do something similar to an integrand of the form $f\left(g'(x)\right) g(x)$, so you can't expect an easy, general rule for integrands of this form.

For example, taking $f(x)=\sin x$ and $g(x) = \ln x$ would lead to $$\int \sin\left(\frac{1}{x}\right) \ln x \, \mbox{d}x$$ which has no elementary anti-derivative.

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