Given the following velocity function of an object moving along a line, find the position function with the given initial position.

\( \int \frac { x ^ { 2 } + 3 x - 12 } { ( x - 2 ) ^ { 2 } ( x + 3 ) ( x ^ { 2 } + 1 ) } d x = \)

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

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

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

Evaluate the given improper integral or show it diverges: \[ \int_{1}^{+\infty} \frac{x^{2}}{\left(x^{3}+1\right)^{2}} d x \]