# Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.2 - Integrals of Vector Functions; Projectile Motion - Exercises 13.2 - Page 753: 3

$$\left(\frac{\pi+2 \sqrt{2}}{2}\right) \mathbf{j}+2 \mathbf{k}$$

#### 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*}

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