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

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

\( \frac{d}{dx}(5^{1+x^{2}+\frac{1}{36}x^{6}-\frac{1}{3}x^{4}}) \)

Let \( f:[a, b] \rightarrow \mathbb{R} \). Show that \( f \notin \mathcal{R}[a, b] \) if and only if there exists \( \varepsilon_{0}>0 \) such that for every \( n \in \mathbb{N} \) there exist tagged partitions \( \dot{\mathcal{P}}_{n} \) and \( \dot{\mathcal{Q}}_{n} \) with \( \left\|\dot{\mathcal{P}}_{n}\right\|<1 / n \) and \( \left\|\dot{\mathcal{Q}}_{n}\right\|<1 / n \) such that \( \left|S\left(f ; \dot{\mathcal{P}}_{n}\right)-S\left(f ; \dot{\mathcal{Q}}_{n}\right)\right| \geq \varepsilon_{0} \).

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

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