$\int_1^2F(x) dx$ and $F'$ of $F(x)= \begin{pmatrix} x^3+5\\ \sin x + \frac{x^2}{x+1}\\ x\exp(x) \end{pmatrix} $

Question

$F:[1,2] \rightarrow \mathbb{R}^3$

$$F(x)= \begin{pmatrix} x^3+5\\ \sin x + \frac{x^2}{x+1}\\ x\exp(x) \end{pmatrix} $$

How can I calculate $\int_1^2F(x) dx$ and $F'$?

Can I just integrate and derive component-wisely?