We have been given a vector valued function:
[tex]sec^{2}(t)i+t(t^{2}+1)^{3}j+t^{2}ln(t)k[/tex]
In order to evaluate the integral of this vector valued function, we will integrate each component of the vector valued function.
[tex](\int sec^{2}(t)dt)i+(\int t(t^{2}+1)^{3}dt)j+(\int t^{2}ln(t)dt)k[/tex]
Upon integrating each of the components, we get:
[tex](\int sec^{2}(t)dt)i+(\int t(t^{2}+1)^{3}dt)j+(\int t^{2}ln(t)dt)k\\\\(tan(t)+c_{1})i+(\frac{(t^{2}+1)^{4}}{8}+c_{2})j+(\frac{t^3(3ln(t)-1)}{9}+c_{3})k[/tex]
