A particle moves along the $x$-axis. The velocity of the particle at time $t$ is given by $v(t)$, and the acceleration of the particle at time $t$ is given by $a(t)$. Which of the following gives the average velocity of the particle from $t=0$ to time $t=5$ ?

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The average value of a function $y$ is given by:

$$\frac{1}{b-a}\int\limits_a^b f(x)   dx$$

For the velocity:

$$\frac{1}{5-0} \int \limits_0^5 v(t)   dt$$ $$= \boxed{\frac{1}{5}\int\limits_0^5 v(t)   dt }$$

$\frac{1}{5} \int\limits_0^5 |v(t)|   dt$ refers to the average speed of the particle. Speed uses the magnitude of the velocity and disregards the direction.