# integration – Explaining a solution to a calculation problem.

I tried to solve a calculation problem and got the correct result, but I don't understand the solution provided at the end of the exercise. Although I got the same answer, I would also like to understand what is happening in the given solution.

Consider the function: $$f (x) = begin {cases} x ^ 2 + ax + b & x leq 0 \ x-1 & x> 0 \ end {cases}$$ Find the antiderivatives of the function $$f$$ if they exist

The solution provided is more or less like this:

by $$f$$ have the function antiderivative $$f$$ must have the darboux
property. (… Some calculations …), therefore $$f$$ has the darboux
property yes and only yes $$b = -1$$ (I understood that now the function
it is continuous, therefore it has an anitederivativa). Using the
consequences of Lagrange's theorem at intervals $$(- infty, 0)$$ Y
$$(0, infty)$$ any antiderivative $$F: mathbb {R} rightarrow mathbb {R}$$ from $$f$$ It has the form:

$$F (x) = begin {cases} dfrac {x ^ 3} {3} + a dfrac {x ^ 2} {2} – x + c_1 & x < 0 \ c_2 & x=0 \ dfrac{x^2}{2} - x + c_3 & x>0 end {cases}$$

$$F$$ being differentiable, it is also continuous, so $$F (0) = c_2 = c_1 = c_3$$.

Therefore, the antiderivatives of $$f$$ have the form:

$$F (x) = c + begin {cases} dfrac {x ^ 3} {3} + a dfrac {x ^ 2} {2} – x & x leq 0 \ dfrac {x ^ 2} {2} – x & x> 0 end {cases}$$

Again, I got the same result, but I don't understand much of the work done previously.

The first thing I didn't understand is the part where they say that $$f$$ It has an antiderivative if it has the Darboux property. I searched a little online and discovered that a function accepts antiderivatives only if it has the Darboux property. So I guess I have to accept that as a fact.

The second (and most important) that I didn't understand was the part where they said they used the consequences of Lagrange's Theorem at intervals. $$(- infty, 0)$$ Y $$(0, infty)$$ to find that first form of antiderivative. What theorem do they refer to? How did they use it at those intervals? Why is there a separate case for $$x = 0$$ with an additional constant $$c_2$$. I just used $$2$$ constants, why they were necessary $$3$$? In a nutshell, I don't understand how they got to that first form of antiderivative and how they used these "consequences of Langrange's theorem." I understood the second form of the antiderivative, that is what I also got, but the first form left me in the dark.

I know these are just details, but I really want to understand what was used here, why it was used and how it was used.