Exercise 38

For each given function $\vr$, determine $\int \vr(t) \ dt$. In addition, recalling the Fundamental Theorem of Calculus for functions of a single variable, also evaluate $\int_0^1 \vr(t) \ dt$ for each given function $\vr$. For each of your calculations, you should write a sentence that states whether your result is a vector or a scalar and describes what is being measured.

1. $\vr(t) = \left\langle \cos(t), \frac{1}{t+1}, te^t \right\rangle$

2. $\vr(t) = \left\langle \cos(3t), \sin(2t), t \right\rangle$

3. $\vr(t) = \left\langle \frac{t}{1+t^2}, te^{t^2}, \frac{1}{1+t^2} \right\rangle$