Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.2 - Integrals of Vector Functions; Projectile Motion - Exercises 13.2 - Page 753: 6

$$\pi \mathbf{i}+\frac{\pi \sqrt{3}}{4} \mathbf{k}$$

We evaluate the integral of the vector function as follows: \begin{align*} \int_{0}^{1}\left(\frac{2}{\sqrt{1-t^{2}}} \mathbf{i}+\frac{\sqrt{3}}{1+t^{2}} \mathbf{k}\right) d t &= 2 \sin ^{-1} t \bigg|_{0}^{1} \mathbf{i}+ \sqrt{3} \tan ^{-1} t \bigg|_{0}^{1} \mathbf{k}\\ &=\pi \mathbf{i}+\frac{\pi \sqrt{3}}{4} \mathbf{k} \end{align*}