Describe the process of sketching a slope field for the differential equation \( y' = x + y \).

\( \begin{array}{ll}\text { (2) } \int \frac{d x}{7-3 x} & \text { (3) } \int \sqrt[3]{x+5} d x \\ \text { (5) } \int \frac{d x}{(2 x-5)^{2}} & \text { (6) } \int \sqrt[3]{(5-3 x)^{2}} d x\end{array} \)

Using Fundamental Theorem of Calculus: Part One, find the following: 1. Given \( f(t)=\left(2+t^{4}\right)^{5} \) and \( g(x)=\int_{1}^{x}\left(2+t^{4}\right)^{5} d t \), what is \( g^{\prime}(x)= \) ?

\[ \frac{n!}{n^{n}} \leq \frac{1}{2^{n-1}}, \quad \forall n \in \mathbb{N}, \quad \text { dopo aver dimostrato per induzione la validità della disuguaglianza } \] verificare che la successione \( a_{n}=\frac{n!}{n^{n}} \) converge a zero.

\( \frac { ( n + 1 ) ! } { ( n + 1 ) ^ { n + 1 } } = \frac { n ! ( n + 1 ) } { ( n + 1 ) ^ { n } ( n + 1 ) } = \frac { n ! } { n ^ { n } ( \frac { n + 1 } { n } ) ^ { n } } = \frac { n ! } { n ^ { n } ( 1 + \frac { 1 } { n } ) ^ { n } } \)

Let \( y=x^{x}+\cos (x) \). Find \( y^{\prime} \).