Question

Evaluate the integrals:

Short answer

$\left[2e^2-e^2\right]i+\left[2\ln 2-2\right]j+\left[2{\tan }^{-1}2-\ln \sqrt{5}\right]-j+\left(\frac{\mathrm{\pi }}{4}-\ln \sqrt{2}\right)k$

Step-by-step solution

Given Information

Consider

The objective is to evaluate the definite integral from $1$and $2$.

The integral of a vector function:

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

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,

Expression

Now,