- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.4 Exercises
- Compute the vector-valued integral:$$\int_0^{\pi/4} (\sin t, \cos t, \tan t)\,dt$$
- Evaluating the integrals component-wise, we get:$$\displaylines{\left(-\text{cos}\,t,\ \text{sin}\,t,-\text{log}\,\text{cos}\,t\right)\,\Biggr|_0^{\pi/4} &= \left(1 - \frac{\sqrt{2}}{2},\ \frac{\sqrt{2}}{2},\ -\text{log}\,\frac{\sqrt{2}}{2}\right)\\ &= \left(1 - \frac{\sqrt{2}}{2},\ \frac{\sqrt{2}}{2},\ \frac{\text{log}\,{2}}{2}\right)}$$Note: the back-of-book solution contains a typo for the rightmost element, which says $\text{log}\,\frac{\sqrt{2}}{2}$ instead of $-\text{log}\,\frac{\sqrt{2}}{2}$ or $\frac{\text{log}\,{2}}{2}.$