Hi I'm asking this in context of this:
Suppose we have non-elementary integral $\chi(x) = \int K(x)dx$ and we want to find
$$\int^b_aK(x)dx$$
Now lets say there is some function $F(x)$ which has elementary integral $\int F(x)$
such that $K(x) + F(x) $ has elementary integral so we can do this :
$$\int^b_aK(x)dx = \int^b_a(K(x)+0)dx$$
$$ = \int^b_a(K(x)+ F(x) - F(x))dx$$
$$=\int^b_aK(x)+ F(x)dx + \int^b_a-F(x)dx$$
Maybe $\int^b_a K(x) + F(x)dx$ is not elementary but lets say I dilute this condition to there exist some definite integral property transformation or substitution which allows us to arrive at numerical value for e.g just like $\int \ln(\sin(x))dx$ which is non-elementary according to WolframAlpha but it gives into some computable value for appropriate limits like
$\int^{\pi/2}_0 \ln(\sin(x))dx$
So my end goal goal is converting $\int^b_a K(x) + F(x)dx$ into some function like that atleast.
I'm not familiar with the deeper details of definite integration. I only want to show that one of the assumptions in your question is wrong and that there are at least some cases (maybe single cases) where a definite non-elementary integral can be evaluated without special functions.
1.) The set of all elementary functions is closed regarding addition (It is an algebraically closed field.). $\int K(x)+F(x)\ dx=\int K(x)\ dx+\int F(x)\ dx$ cannot be elementary therefore. Your prerequisite "such that $K(x)+F(x)$ has elementary integral" is wrong therefore.
2.) Definite integrals can be calculated also by other methods than indefinite integration.
Lichtblau, D.: Symbolic definite integration: methods and open issues. ACM Communications in Computer Algebra 45 (2011) (1/2) 1-16
Raab, C. G.: Definite Integration in Differential Fields. PhD thesis Johannes Kepler University Linz, Austria, 2012
Algorithms for symbolic definite integration?
Davenport, G.: An Exploration of Three Related Parametric Definite Integrals. 2016
Davenport, G.: The Difficulties of Definite Integration
A non-elementary definite integral and therefore a sum of an elementary and a non-elementary definite integral can therefore be evaluated in some cases without special functions.
Nyblom, M. A.: On the evaluation of a definite integral involving nested square root functions. Rocky Mountain J. Math. 37 (2007) 4 1301-1304
3.) One could ask: Is there a theory of integration in elementary terms for definite integrals? But it seems there is no such theory.
4.) The value of a definite integral is a number. One could ask if this number can be an elementary number. But that's a completely different mathematical problem.