Q: Application of Fundamental Theorem of Calculus, splitting integrals Looking for some help and explanation on some questions that require FTC.
I will be using the following versions of FTC. I have made an attempt but they do not correspond to the solutions I was given out so am looking for feedback on why my method is incorrect.
FTC 1
Let $f$ be regulated on the interval $[a,b]$ and $f$ is continuous on at $c \in (a,b)$. Then $F(x) = \int^x_a f$ is differentiable on $(a,b)$ with $F'(c) = f(c)$ 
FTC 2
Let $f: [a,b] \rightarrow \mathbb{R}$ be continuous and let $g: [a,b] \rightarrow \mathbb{R}$ be differentiable with $g'(x) = f(x) \forall x \in [a,b]$. Then $\int^b_a f = g(b) - g(a)$ 
Questions
1) Let $I = \int^b_a \frac{1}{log x}dx$ 
Find: $\frac{dI}{da}, \frac{dI}{db}$ 
So my method is to let $F$ be such that $F' = \frac{1}{log x}$, and then using FTC2, I get $I = F(b) - F(a)$. After differentiation I get $\frac{1}{\log a}, -\frac{1}{\log b}$ respectively.
Now the solutions suggest that I split the integral up into $\int^b_2, \int^2_a =\int^b_2 - \int^a_2$, and then differentiate here. Am I missing something about the conditions? Do I need to rearrange each equation into the indefinite integral?
2) Same as above except now $I = \int^{b(x)}_{a(x)} \frac{1}{\log x} dx$ I need to find $I'(x)$ $a(x)$ and $b(x)$ are functions with values in $[2, \infty)$ 
In the form of b(x), a(x) I can't seem to rearrange it into the indefinite integral form. From the solutions they still seem to recommend that I write
$I(x) = J(a(x)) - J(b(x))$ where $J(a) = \int^a_2 \frac{1}{log t} dt$ and then differentiating with the chain rule. I don't understand why this is necessary, as to me, an unrigorous solution that uses the same idea $\frac{d}{dx}(F(b(x)) - F(a(x))$ does the same thing, so why do we need to split the integral like this?
3) Can someone give me an example of how they would split $\int^{1+x^2}_{x^2}f(t) dt, x \in \mathbb{R}$ 
4) Where do I need to comment about differentiability / continuity? Do I need to prove that the functions I here are continuous before I use the theorem?

A: I don't know what you mean by "indefinite integral form" because these are all definite integrals.  Any integral with limits of integration is definite.  No limits of integration means the integral is indefinite.
Example of indefinite integral:  $\displaystyle \int \frac{1}{\log x} \, dx$
Example of definite integral:  $\displaystyle \int_a^b \frac{1}{\log x} \, dx$

To loosely paraphrase the FTC1:
$$ \frac{d}{dx} \int_a^x f(t) \, dt = f(x),$$
where $a$ is any constant.
So when you want to apply FTC1, you should make sure your integral has this form.  Sometimes you'll need to split the integral up into the sum of integrals.  See my answer for your #3 below for an example.

Your question #1:
So for $I = \displaystyle \int_a^b \frac{1}{\log x} \, dx$, we have
$$ \frac{d}{db} \int_a^b \frac{1}{\log x} \, dx = \frac{1}{\log b}$$
and we have
\begin{align}
  \frac{d}{da} \int_a^b \frac{1}{\log x} \, dx
    &= \frac{d}{da} \left(-\int_b^a \frac{1}{\log x} \, dx\right)\\[0.3cm]
    &= -\frac{d}{da} \int_b^a \frac{1}{\log x} \, dx\\[0.3cm]
    &= -\frac{1}{\log a}
\end{align}

Your question #2:
If I'm understanding you correctly, then you are correct about how it's the same idea as $\dfrac{d}{dx}\left(F(b(x)) - F(a(x))\right)$.  But the FTC1 explicitly only covers integrals of the form $\displaystyle \int_a^x f(t) \, dt$, or, more generally, $\displaystyle \int_a^{b(x)} f(t) \, dt$.  Splitting the integral up (like I did with #3 below) or swapping the limits (like I did with #1 above) is essentially just a matter of rigorously showing that the FTC1 can actually be applied.

Your question #3:
\begin{align}
  \int_x^{1+x^2} f(t) \, dt &= \int_x^0 f(t) \, dt + \int_0^{1+x^2} f(t) \, dt\\[0.3cm]
    &= -\int_0^x f(t) \, dt + \int_0^{1+x^2} f(t) \, dt
\end{align}
Now both of these integrals have the form required to use FTC1.

Your question #4:
It depends on your instructor, but I'd be astonished if somewhere out there is a Calculus I (or II) class that requires students to prove continuity or differentiability before using theorems.  The most I can reasonably imagine is that your instructor would want you to at least simply state that the function is continuous/differentiable/satisfies these hypotheses/etc. before proceeding.
