Evaluate the integral:

Consider ${\int }_0^{2\mathrm{\pi }}(\sin t,\cos t,t)dt$

The objective is to evaluate the definite integral from $0$to $2\mathrm{\pi }$.

The integral of a vector function

$$\begin{gathered} \int r\left(t\right)dt=\int x\left(t\right)idt+\int y\left(t\right)jdt+\int z\left(t\right)kdt \\ =i\int x\left(t\right)dt+j\int y\left(t\right)dt+k\int z\left(t\right)dt+c \end{gathered}$$

Where $c$is a vector constant and has the form localid="1649681269368" $c=\left(c_1,c_2,c_3\right)$for scalar constants localid="1649681273626" $c_1,c_2$and localid="1649681280075" $c_3$.

Now,

localid="1649681285595"

The anti derivative of is . To compute the definite integral, the fundamental theorem of calculus is used. It states thatlocalid="1650008560018" ${\int }_a^{b}f\left(x\right)dx=F\left(b\right)-F\left(a\right)$where$F$is the anti derivative of$f$.

Now,

Thus