Can Integration by Parts be used to define the explicit inverse or integral-inverse of a function defined by an integral?

Question

1. Suppose you have a function $f(x)$ defined as an integral, such that $$f(x)=\int g(x)dx$$ like say the natural logarithm defined as $$\ln(x)= \int_{1}^{x} \frac{1}{t}dt.$$
   
   
   

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2. Unfortunately, the inverse function theorem, which is $$f^{-1}(x)'= \frac{1}{f'(f^{-1}(x))}$$ can at best only be arranged into this form after integrating both sides: $$f^{-1}(x)+C = \int \frac{dx}{f'(f^{-1}(x))}$$ As you can see, the inverse of the original function appears on both sides of the equation, so this is not in any way helpful for finding the unknown explicit inverse of a particular function defined as an integral unless you somehow already know the answer to begin with.
   
   
   

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3. So, I am wondering if there is a way to manipulate intergation by parts to find the inverse of a function defined as an integral. Starting the statement of the formula, $$ \int u(x) v'(x)dx = u(x)v(x) - \int v(x) u'(x)dx$$ we can see the generalization of the integral of simply a single function is $$ \int g(x)dx = g(x)x - \int x g'(x)dx$$ However, $x$ can also be assumed as $x=g^{-1}(g(x))$ which, when substituted, would make the equation $$ \int g(x)dx = g(x)g^{-1}(g(x)) - \int g^{-1}(g(x)) g'(x)dx.$$
   
   
   

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4. Now for me, this is where I'm not completely sure, because if you look at the second term of the right side of the equation, that looks an awful lot like an instance of the chain rule for differentiation. It would seem the best arrangement of the equation is $$ \int g(x)dx = g(x)g^{-1}(g(x)) - \int g^{-1}(g(x)) g'(x)dx$$ $$ \int g(x)dx = g(x)x - \int g^{-1}(x)dx$$ For some reason, this doesn't seem completely correct to me, and I don't know how to take it further to accomplish the goal. Because, if you integrate $e^x$ for instance, the integral is still just $e^x$, and, $$ \int e^x dx \neq e^x*x- x(\ln(x)-1)$$ But, if I finished the integral, it still doesn't work, $$ \int g(x)dx = g(x)x - g^{-1}(x)-C$$ $$\int e^x \neq e^x x - \ln(x)$$
   Can this method be salvaged so that one can define the explicit inverse of a function defined as an integral?

Answer 1

Ok here is the main mistake in your argument. The equation $$\int g(x) \, dx=xg(x) - \int g^{-1}(x)\,dx\tag{1}$$ does not hold and the reason is that the following equation does not hold $$\int g^{-1}(g(x))g'(x)\,dx=\int g^{-1}(x)\,dx\tag{2}$$ That these equations don't hold can be easily checked by differentiating both sides of the equation. Perhaps you are trying to use substitution in integrals (which is more of a restatement of chain rule of derivatives) but not exactly using the new variable. The correct way to write the equation $(2)$ is as follows $$\int g^{-1}(g(x))g'(x)\,dx=\int g^{-1}(u)\,du\tag{3}$$ where $u=g(x) $.