I recently was looking at the derivation for $\Gamma(x)$'s approximation to the factorial function and greatly appreciated one of the ways to solve it by considering an integral $\int_0 ^\infty e^{ay}dy=\frac{1}{a}$ and differentiating both sides $n$ times, and substituting $a=-1$ to obtain the result.
I was inspired by this method to try and use it to solve $\int \ln(x) dx$, however I am unaware whether the method actually is correct since it is very 'handwavy'. All I know is that it matches with the true formula for the integral. Here is the procedure:
Use the substitution $y=\ln x$ to obtain $\int y e^y dy$ and consider a function $f(y,a)=e^{ay}$ then the integral is just $\int \frac{\partial f}{\partial a}(y,1)dy$.
Switching the integral and derivative operators (I am aware of some conditions necessary for this to hold) to obtain: $$\frac{\partial}{\partial a}\int f(y,a)|_{a=1} dy$$ $$=\frac{\partial}{\partial a}\int f(y,a)|_{a=1} dy$$ $$=\frac{\partial}{\partial a} [\frac{e^{ay}}{a}+c(a)]|_{a=1}$$ $$(-\frac{e^{ay}}{a^2}+\frac{ye^{ay}}{a}+c'(a))|_{a=1}$$ $$=-e^{y}+ye^{y}+c'(1)=$$
Converting the substitution back, $$=x\ln x-x+c'(a)|_{a=1}$$
Let me know your thoughts on where I have made a mistake or what could be corrected :) I look forward to your comments.
In finding an antiderivative, all sorts of dubious formal manipulations are legitimate. The final result can always be checked by differentiation. It is only for definite integrals that you have to be careful.