I search an alternate expression for
∫f(x)g(x)
I'd like to express it in terms of $f$ and $g$ and $n$-th derivatives and antiderivatives of $f$ and $g$.
For example, I could use something like this: (if it where a solution)
f'(x)+F(x)g(x)-5g'(x)x+ln(x)
I don't mind if those terms are "of infinite length". (like for example a Taylor series)
However, I would prefer a "finite term" as solution. (regardless of its length)
I doubt that there is a solution to this in form of a "finite term", using only $f$ and $g$ and $n$th derivatives and antiderivatives of $f$ and $g$, because that would render the partial integration quite useless.
I cannot use solutions containing any Integrals over "composite terms".
like, for example:
∫f(x)g(x)dx=F(x)g(x)-∫F(x)g'(x)dx
If the solution uses a Taylor series, I'd like to know when exactly this solution can be employed, because as far as I know, a taylor series does not always converge.
similarly, if the solution uses any other infinite series approaching the result, Id like to know under wich circumstances it can be applied, too. (are there any cases where the series used does not converge?)
The only known solution to your problem is integration by parts and you do not like it .
Other than that you need restrictions on $f$ and $g$ to try substitution .
In special cases such as polynomials or some trig functions you may find answers but in general the problem is too broad to be solved.