\( \int _ { - 1 } ^ { 1 } \frac { x \cos ^ { 2 } x } { 1 + x ^ { 2 } } d x \)

) 8. Al sustituir \( u=\operatorname{sen} \theta \), ¿qué resultado se obtiene al evaluar \( \int \operatorname{sen}^{2} \theta \cos \theta d \theta \) ? \( \begin{array}{llll}\text { A) }-\cos ^{2} \theta \operatorname{sen} \theta+C & \text { B) } \frac{1}{6} \operatorname{sen}^{3} \theta \cos ^{2} \theta+C & \text { C) } 2 \operatorname{sen}^{2} \theta \cos ^{2} \theta-\operatorname{sen}^{3} \theta & \text { D) } \frac{1}{3}\end{array} \)

12) [10 points] Determine which of the following definite integrals equals \( \lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{n} \sqrt{\frac{i}{n}} \).

Given the differential equation \( \frac{dy}{dx} = 3y \) and the initial condition \( y(0) = 2 \), find the particular solution.

\[ f(x)=3 x^{x^{4}} \] Calcule el valor, redondeando a cuatro decimales, de \( f^{\prime}(x) \) en el punto \( x=1,07 \).

\( y = ( \frac { 1 + e ^ { 2 x } } { 1 - e ^ { 2 x } } ) \operatorname { Ln } ( \sqrt { 4 x + 5 } ) \)