Answer
$$\int\left(\frac{1}{t} +\mathbf{j}-t^{3 / 2} \mathbf{k}\right) d t=\ln |t| \mathbf{i}+t \mathbf{j}-\frac{2}{5} t^{5 / 2} \mathbf{k}+\mathbf{C}$$
Work Step by Step
Given $$\int\left(\frac{1}{t} +\mathbf{j}-t^{3 / 2} \mathbf{k}\right) d t $$
Integrating on a component-by-component basis produces
\begin{align}\int\left(\frac{1}{t} +\mathbf{j}-t^{3 / 2} \mathbf{k}\right) d t&=\ln |t| \mathbf{i}+t \mathbf{j}-\frac{1}{5/2} t^{\frac{3}{2}+1} \mathbf{k}+\mathbf{C}\\
&=\ln |t| \mathbf{i}+t \mathbf{j}-\frac{2}{5} t^{5 / 2} \mathbf{k}+\mathbf{C}
\end{align}
