Un físico mide la velocidad de una partícula en función del tiempo, dada por \( v(\mathrm{t})=\mathrm{t}^{3}-3 \mathrm{t}+2 \). Calcula la distancia recorrida por la partícula en el intervalo [1,2]. La distancia total recorrida en ese intervalo es a. 5 b. 4.5 c. 5.5 d. 6

Find the derivative of \( f(w)=\frac{3}{\left(w^{2}-6\right)^{5}} \)

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

Residues at Poles - Example 1 The function \( \quad f(z)=\frac{z+1}{z^{2}+9} \) has an isolated singular point at \( \mathrm{z}=3 \mathrm{i} \) and can be written

(c) \( \int_{6}^{0}(2+5 x) e^{\frac{1}{3} x} d x \)

Residues at Poles = Proof the Theorem Assume \( f(z) \) has the following form \[ f(z)=\frac{\phi(z)}{\left(z-z_{0}\right)^{m}} \] where \( \varphi(\mathrm{z}) \) is analytic and nonzero at \( \mathrm{z}_{0} \), then it has Taylor series representation