Question

Evaluate the integral:

Short answer

Step-by-step solution

Given Information

Consider

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

Calculation

The integral of a Vector Function

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

Where $c$is a vector constant and has the form $c=(c_1,c_2,c_3)$for scalar constants $c_1,c_2$and $c_3$.

Now,

$$\begin{gathered} \int ||\sin t,\cos t,t||dt \\ =\int \sqrt{{\sin }^{2}t+{\cos }^{2}t+t^2}dt \\ =\int \sqrt{1+t^2}dt \\ =\frac{t}{2}\sqrt{1+t^2}+\frac{1}{2}\sin h^{-1}\left(t\right) \end{gathered}$$

Expression

Now,

Thus