Q2. Find the equation for the curve that is an extremal for the functional \[ \begin{array}{l}f(x)=\int_{0}^{t}\left[t \dot{x}(t)+\dot{x}^{2}(t)\right] d t \\ \text { Boundary Conditions are } x(0)=1, t_{f}=2 \text { and } x(2) \text { is free. }\end{array} \]

Sketch the region enclosed by \( y=e^{2 x}, y=e^{4 x} \), and \( x=1 \). Find the area of the region.

Realiza la investigación encome Integrar: de resolver los ejercicios: 1. \( \int x^{3} \cdot e^{x} d x \) 2. \( \int|1-x| \cdot \sqrt{x} d x \) 3. \( \int \frac{3 x}{\sqrt[3]{x^{2}+3}} d x \)

\( \int \sqrt[ \infty ] { x ^ { r } } d x \)

A factory selling cell phones has a marginal cost function \( C(x)=0.01 x^{2}-3 x+229 \), where \( x \) represents the number of cell phones, and a marginal revenue function given by \( R(x)=429-2 x \). Find the area between the graphs of these curves and \( x=0 \).

Evaluate the \( \lim _{x \rightarrow-2}\left(\frac{x^{3}+6 x^{2}+8 x}{x+2}\right) \) Solution: