In apostol chapter on fundamental theorems of calculus it is suggested to validate the following equation:
$$\int_0^x (t + |t|)^2 dt = \frac{2x^2}{3}(x + |x|),x \in \mathcal{R}$$
Of course, this can be solved by reviewing 3 cases $x <=> 0$. However, I thought that this exercise should use the fundamental theorems somehow. So I did it in reverse order:
Split the real line into 2 intervals: $x \geq 0$ and $x < 0$. Now, differentiate the right side piecewise. Since the function $f(x) = (x + |x|)^2$ is continuous everywhere, by the 1st fundamental theorem of calculus, the function $A(x) = \frac{2x^2}{3}(x + |x|)$ is the antiderivative of $f(x)$. Now, since $A(0) = 0$, by the second theorem of calculus, the integral on the left is equal to the right hand side as $A(x) - A(0)$.
However, the 2nd theorem of calculus has a condition: the function $f(x)$ should have the antiderivative on the same OPEN interval as it is defined on. And the point $x = 0$ should be the interior point (Apostol theorem 5.3)! I thought this solution works, but the fact is: the integral $A(x)$ has the lower bound as $x = 0$, which is not within the interval, where $f(x)$ has antiderivative (since the antiderivative has to be differentiable at 0, and here it is not). How can I change my argument so that I can use the 2nd fundamental theorem of calculus here? Or explain what is wrong with my reasoning.