- 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:$$\int_0^1 \left(\frac{e^t}{1+e^t}\mathbf{i} + \frac{1}{1+e^t}\mathbf{j}\right)\,dt$$
- Use a $u$ substitution where $u(t) = 1 + e^t$ and $du = e^t\, dt.$ Then, rewrite $\frac{1}{1 + e^t} = 1 - \frac{e^t}{1 + e^t}$
- Evaluating the integral, we get:$$\displaylines{\int_0^1 \left(\frac{e^t}{1+e^t}\mathbf{i} + \frac{1}{1+e^t}\mathbf{j}\right)\,dt &= \text{log}\,(1 + e^t)\mathbf{i} + t - \log(1 + e^t)\,\mathbf{j}\,\Biggr|_0^1\\ &= \text{log}\frac{1 + e}{2}\mathbf{i} + \left(1 - \text{log}\frac{1 + e}{2}\right)\mathbf{j}}$$