Answer
$$\left(\frac{\pi+2 \sqrt{2}}{2}\right) \mathbf{j}+2 \mathbf{k}$$
Work Step by Step
Given $$\int_{-\pi / 4}^{\pi / 4}\left[(\sin t) \mathbf{i}+(1+\cos t) \mathbf{j}+\left(\sec ^{2} t\right) \mathbf{k}\right] d t $$
Then
\begin{align*}
\int_{-\pi / 4}^{\pi / 4}\left[(\sin t) \mathbf{i}+(1+\cos t) \mathbf{j}+\left(\sec ^{2} t\right) \mathbf{k}\right] d t&=[-\cos t]\bigg|_{-\pi / 4}^{\pi / 4} \mathbf{i}+[t+\sin t]\bigg|_{-\pi / 4}^{\pi / 4} \mathbf{j}+[\tan t]\bigg|_{-\pi / 4}^{\pi / 4} \mathbf{k}\\
&=\left(\frac{\pi+2 \sqrt{2}}{2}\right) \mathbf{j}+2 \mathbf{k}
\end{align*}
