Fundamental Theorem of Calculus

Need some help with FTC problems. My calculus textbook defines the FTC in two parts:

Definition Part 1:

If $\displaystyle f$ is a continuous function on $\displaystyle [a,b]$, then the function $\displaystyle g$ defined by $\displaystyle g(x) = \int_a^x f(t)dt$ where $\displaystyle a\leq x\leq b$

is continous on $\displaystyle [a.b]$ and differentiable on $\displaystyle (a,b)$, and $\displaystyle g'(x) = f(x)$

Definition Part 2:

If $\displaystyle f$ is a continuous function on $\displaystyle [a,b]$, then $\displaystyle \int_a^b f(x)dx=F(b)-F(a)$ where F is any antiderivative of $\displaystyle f$, that is, a function such that $\displaystyle F'=f$

Whew, here are the problems I'm confused on...

Problem 1. Using Part 1 definition, find the derivative of the function:

$\displaystyle y=\int_{1-3x}^1 \frac{u^3}{1+u^2}du$

Problem 2. Using Part 2 defintion, evaluate the integral:

$\displaystyle \int_0^2 x(2+x^5)dx$

Thanks a lot.