Calculate the area under the curve defined by the function \( f(x) = x^3 + 2x - 1 \) from \( x = 1 \) to \( x = 3 \).

3. Sia \( f(x)=x^{7} e^{3 x} \). Allora si ha \( \begin{array}{l}\text { (a) } f^{(8)}(0)=0 \\ \text { (b) } f^{(7)}(0)=0 \\ \text { (c) } f^{(8)}(0)=3 \cdot 8! \\ \text { (d) } f^{(3)}(0)=8 \cdot 3! \\ \text { (e) } f^{(3)}(0)=24\end{array} \)

Determina si la integral impropia \( \int_{0}^{1} \frac{1}{\sqrt{x}} \, dx \) converge o diverge.

La funzione \( f(x)=\sin \left(x^{4}\right)-\sinh \left(x^{4}\right) \) soddisfa \( \begin{array}{l}\text { (a) } f(x) \geq 0 \text { in un intorno di } x=0 \\ \text { (b) } f(x) \leq 0 \text { in un intorno di } x=0 \\ \text { (c) } f(x) \sim x^{4} \text { per } x \rightarrow 0 \\ \text { (d) } f(x)=0\left(x^{12}\right) \text { per } x \rightarrow 0 \\ \text { (e) nessuna delle altre risposte }\end{array} \)

4. \( [3+2=5 \) points \( ] \) (a) Find the area of the region enclosed by: \[ y=\sin x, \quad \text { and } \quad y=2 \cos x, \quad \text { over the interval }[0, \pi] \]

(2) \( \int \frac{10 x^{2}+6 x+4}{2 x+2} d x \)