Use Newton's method to find an approximate solution to \( x^{2}-2=0 \), starting from an initia estimate \( x_{1}=2 \). Iterate until 6 iterations. Hints: \( \boldsymbol{f}^{\prime}(\boldsymbol{x})=2 \boldsymbol{x} \) Marks)

\( y ^ { \prime } ( x ) = \frac { ( u - 3 ) ^ { 2 } } { u ^ { 2 } - 7 u + 10 } \)

5) \( \int \frac{24 x^{2}-8 x}{2 x^{3}-x^{2}} d x \)

Ejercicio 3. Calcular \[ \begin{array}{ll}\text { i) } \int_{1}^{2}(4 x-2) e^{3 x+1} \mathrm{~d} x & \text { ii) } \int \frac{\cos (x)}{\operatorname{sen}(x)-\operatorname{sen}^{3}(x)} \mathrm{d} x\end{array} \quad \text { iii) } \int_{2}^{e} \frac{\sqrt{\ln (x)+2}}{x} \mathrm{~d} x \]

2) \( \lim _{x \rightarrow 0}\left(\frac{1}{x \sqrt{1+x}}-\frac{1}{x}\right)= \)

Знайти площу фігури, обмеж а) \( y=4 x-x^{2} ; \)