- 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:$$\int_0^1 (te^t\mathbf{i} + t^2e^t\mathbf{j} + te^{-t}\,\mathbf{k})\,dt$$
- Use integration by parts with $u = t^n$ and $dv = e^t\, dt$
- Using integration by parts on each component, we get:$$\left(te^t - e^t\right)\Biggr|_0^1\textbf{i}\ + \left(t^2e^t - 2(te^t - e^t)\right)\Biggr|_0^1\textbf{j}\ + \left(-te^{-t} - e^{-t}\right)\Biggr|_0^1\textbf{k}$$Evaluating over the limits of integration, we get:$$\left(1,\ e-2,\ 1 - \frac{2}{e}\right)$$