| Section | You should be able to… | Examples | Review Exercises |
|---|---|---|---|
| 7.1 | 1 Integrate by parts (p. 472) | 1–6 | 8, 9, 16, 19, 21, 25, 47, 48 |
| 2 Derive a formula using integration by parts (p. 476) | 7, 8 | 37, 38 | |
| 7.2 | 1 Find integrals of the form \(\int \sin ^{n}x\,dx\) or \(\int \cos ^{n}\!x\,dx,\) \(n \ge 2\) an integer (p. 480) | 1–3 | 5, 31 |
| 2 Find integrals of the form \(\int \sin ^{m}x\,\cos^{n}x\,dx\) (p. 483) | 4 | 10, 28 | |
| 3 Find integrals of the form \(\int \tan ^{m}x\,\sec^{n}x\,dx\) or \(\int \cot ^{m}x\,\csc ^{n}x\,dx\) (p. 483) | 5–7 | 3, 4 | |
| 4 Find integrals of the form \(\int \sin (ax) \sin (bx) \,dx\), \( \int \sin (ax) \,\cos (bx)\,dx,\) or \(\int \cos (ax) \,\cos (bx) \,dx\) (p. 485) | 8 | 32, 33 | |
| 7.3 | 1 Find integrals containing \(\sqrt{a^{2}-x^{2}}\) (p. 488) | 1 | 6, 11 |
| 2 Find integrals containing \(\sqrt{x^{2}+a^{2}}\) (p. 489) | 2, 3 | 18, 24, 26 | |
| 3 Find integrals containing \(\sqrt{x^{2}-a^{2}}\) (p. 491) | 4 | 7, 30 | |
| 4 Use trigonometric substitution to find definite integrals (p. 492) | 5, 6 | 34, 35 | |
| 7.4 | 1 Find an integral that contains a quadratic expression (p. 496) | 1–3 | 1, 14, 20 |
| 7.5 | 1 Integrate a rational function whose denominator contains only distinct linear factors (p. 500) | 1, 2 | 12, 27, 29 |
| 2 Integrate a rational function whose denominator contains a repeated linear factor (p. 502) | 3 | 17, 23 | |
| 3 Integrate a rational function whose denominator contains a distinct irreducible quadratic factor (p. 503) | 4 | 2, 15 | |
| 4 Integrate a rational function whose denominator contains a repeated irreducible quadratic factor (p. 504) | 5 | 22 | |
| 7.6 | 1 Approximate an integral using the Trapezoidal Rule (p. 508) | 1–5 | 49(a), 50 |
| 2 Approximate an integral using Simpson’s Rule (p. 514) | 6–8 | 49(b) | |
| 7.7 | 1 Use a Table of Integrals (p. 520) | 1–3 | 36(a) |
| 2 Use a computer algebra system (p. 522) | 4 | 36(b) | |
| 7.8 | 1 Find integrals with an infinite limit of integration (p. 524) | 1, 2 | 39, 42, 44 |
| 2 Interpret an improper integral geometrically (p. 525) | 3, 4 | 51, 52 | |
| 3 Integrate functions over \([a,b]\) that are not defined at an endpoint (p. 527) | 5–7 | 40, 41, 43 | |
| 4 Use the Comparison Test for Improper Integrals (p. 529) | 8 | 45, 46 |