Master the Art of FTOC: Comprehensive Examples and Expert Advice

Using the FTOC to find the derivative of the integral
${\int <!--\; \int \; -->}_{x^2}^5(4x+2)dx.$

Is FTOC valid only in interval that the function is continuous?
${\mathrm{\partial <!--\; \partial \; -->}}_x{\int <!--\; \int \; -->}_0^{x}f(x^{\prime })d{x}^{\prime }=f(x)-<!--\; -\; -->f(0)$

Let $f:<!--\; :\; -->[a,b]\to <!--\; \to \; -->\mathbb{R}$ be monotone increasing, $F:<!--\; :\; -->[a,b]\to <!--\; \to \; -->\mathbb{R}$ such that $F(x):={\int <!--\; \int \; -->}_{[a,x]}f$ and $x_0\in <!--\; \in \; -->[a,b]$. Then $F$ is differentiable at $x_0$ if and only if $f$ is continuous at $x_0$

Calculate:
${\int <!--\; \int \; -->}_{-<!--\; -\; -->2}^2\sin <!--\; \; -->(x^5)e^{(x^8\sin <!--\; \; -->(x^4))}dx$

Find the derivative of ${\int <!--\; \int \; -->}_3^x{\sin }^3<!--\; \; -->tdt$

The FTOC states that if $f$ is continuous on $[a,b]$ then it is integrable.
If $f$ is not defined at certain points of $[a,b]$ we can often give meaning to an improper integral. But under what circumstances will $f$ always be integrable on it's domain, properly or improperly?
If we partition $[a,b]$ and consider $f$ on the subintervals $(x_i,x_{i+1})$ and assume it is continuous, (so $f$ is piecewise continuous on $[a,b]$), is $f$ then integrable on the entire interval simply by the adding up each integral over each subinterval? But the interval is open, so how does FTOC and improper integrals come into play? Will the improper integral over $[x_i,x_{i-<!--\; -\; -->1}]$ always be well-defined?

Are there any cases which the First Fundamental Theorem of Calculus would fail?

Use the fundamental theorem of calculus:
Find $F^{\prime }(x)$, where $F(x)={\int <!--\; \int \; -->}_0^{x^3}{\sin }^3<!--\; \; -->tdt$

First fundamental theorem of calculus uses a function
$F(x)={\int <!--\; \int \; -->}_a^{x}f\left(t\right)dt$ for $f$ a continuous function between $[a,b]$ where $x$ is between $[a,b]$
$F(x)$ is an antiderivative of function $f$.
So what closed interval is considered when we take indefinite integral?

Let $f$ be an integrable function on $[a,b]$, $F$ be the antiderivative of $f$. Then
${\int <!--\; \int \; -->}_a^{b}fdu=F(b)-<!--\; -\; -->F(a)$
Is it suggesting that while $f$ is integrable, the derivative of the integral may not be $f$?