It is easy to solve integrals of the form \int\frac{f'}f using the defintion of the natural logarithm: \int \frac{f'(x)}{f(x)}\;\mathrm dx = \ln f(x).\ Is there a similar identity for the case \int\frac f{f'}?
Is there a general approach to solve integrals of the form $\int\frac f{f'}$?
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integration
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1[Interesting ..](http://www.wolframalpha.com/input/?i=integration%28f%2Ff%27+dx%29) – 2012-03-28
1 Answers
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Writing $f = e^g$ we have \int \frac{f}{f'} = \int \frac{1}{g'} and this can be a more or less arbitrary integrand so no. Already taking $g = x \ln x - x$ we get a non-elementary integral.
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2@Mark: $g'$ can be a more or less arbitrary integrand, but $\int g' = g$ is no more complicated than $g$; that is, this construction doesn't "increase the complexity." On the other hand, as the above example shows, $\int \frac{1}{g'}$ can be non-elementary even if $g$ is. – 2012-03-28