Integrating $\int_{}^{} \frac{f'(r)}{f(r)}\frac{1}{1+ln(\frac{a}{f(r)})}dr$

Question

I am trying to find a solution for this integral: $\int_{}^{} \frac{f'(r)}{f(r)}\frac{1}{1+ln(\frac{a}{f(r)})}dr$

My professor hinted that guessing the solution is a good way to go, but I am unable to make the correct guess.

Answer 1

Hint: $$\int \frac{1}{\clubsuit }\cdot \frac{1}{1-\ln \clubsuit } \mathrm{d}\clubsuit =-\ln | 1-\ln\clubsuit| +C $$

end
$$\int \frac{1}{1- \spadesuit } \mathrm{d}\spadesuit =-\ln |1-\spadesuit| +C. $$

Answer 2

Hint: What is the function whose derivative is $\frac{f'(r)}{f(r)}$ ?

Make the substitution \begin{align} u = \ln{\frac{a}{f(r)}} &= \ln{a} - \ln{f(r)} \\ -du &= \frac{f'(r)}{f(r)} dx \end{align}

\begin{align} I &= -\int{\frac{du}{1 + u}} \\ &= -\log{\left|1+u\right|} + C \\ &= -\log{\left| 1 + \ln{\left( \frac{a}{f(r)} \right)} \right|} + C \end{align}