Fundamental theorem of calculus part 2

Question

In the fundamental theorem of calculus part 2 we have $$\int_a^bf(x)dx=F(b)-F(a)$$ where $F(x)$ is any anti derivative of $f(x)$ on $[a, b]$ but i want to know that what will happen if the equality $F'(x)=f(x)$ does not hold at some point in $[a, b]$ or if $F(x)$ is continuous but not differentiable at a point in $[a, b]$

Answer 1

Then you break the limits into two parts. Suppose that point is $p$ Then the new equation would be: $$\int^{y}_{x}f(x)dx=\int^{y}_{p}f(x)dx+\int^{p}_{x}f(x)dx$$

Example: Since, $f(x)=|sin(x)|$ cannot be differentiated at $\pi/2$. So, $$\int^{pi}_{0}|sin(x)|dx=\int^{pi}_{\pi/2}-sin(x)dx+\int^{\pi/2}_{0}sin(x)dx$$