Newton Leibnitz's Theorem

Let $f:\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\to \mathrm{ℝ}$ be a continuous function such that $f\left(0\right)=1$ and ${\int }_0^{\frac{\pi }{3}}f\left(t\right)dt=0$

Then which of the following statements is (are) TRUE?

If $g\left(x\right)={\int }_{\sin x}^{\sin 2x}{\sin }^{-1}tdt,$  then

The value of  $I={\displaystyle \sum }_{k=1}^{98}\underset{k}{\overset{k+1}{{\displaystyle \int }}}\frac{k+1}{x(x+1)}dx,$ then

Let $f:\left[0,2\right]\to R$ be a function which is continuous on $\left[0,2\right]$ and is differentiable on $\left(0,2\right) f\left(0\right)=1$. Let $F\left(x\right)=\underset{0}{\overset{x^2}{\int }}f\mathrm{ }\left(\sqrt{t}\right)dt$ for $x\in \left[0,2\right].$ If $F^{\text{'}}\left(x\right)=f^{\text{'}}\left(x\right)$ for all $x\in \left(0,2\right),$ then $F\left(2\right)$ equals