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.
