X2 t04 05 trig substitutions (2013)

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  • 1. Trig Substitutions

2. Trig Substitutionsa2 x2 use x a tan 3. Trig Substitutionsa2 x2 use x a tan a2 x2 use x a sin or x a cos 4. Trig Substitutionsa2 x2 use x a tan a2 x2 use x a sin or x a cosx2 a2 use x a sec 5. 1e.g. i dx a x2 2 6. 1e.g. i dx a x2 2 7. 1e.g. i dx x a tan a x 2 2 a sec 2 ddx a sec 2 d 2a a 2 tan 2 8. 1e.g. i dxx a tan a x 22 a sec 2 ddx a sec 2 d 2a a 2 tan 2 a sec 2 d 2a sec 2 secd 9. 1e.g. i dx x a tan a x 22 a sec 2 d dx a sec 2 d 2a a 2 tan 2 a sec 2 d 2a sec 2 secd logsec tan c 10. 1e.g. i dx x a tan a x 22 a sec 2 d dx a sec 2 d 2a a 2 tan 2 a sec 2 d 2a2 x2a sec 2 x secd logsec tan ca 11. 1e.g. i dx x a tan a x 22 a sec 2 d dx a sec 2 d 2a a 2 tan 2 a sec 2 d 2a2 x2a sec 2 x secd logsec tan ca a2 x2 x log c aa 12. 1e.g. i dx x a tan a x 22 a sec 2 d dx a sec 2 d 2a a 2 tan 2 a sec 2 d 2a2 x2a sec 2 x secd logsec tan ca a2 x2 x log c aa log a 2 x 2 x log a c 13. 1e.g. i dx x a tan a x 22 a sec 2 d dx a sec 2 d 2a a 2 tan 2 a sec 2 d 2a2 x2a sec 2 x secd logsec tan ca a2 x2 x log c aa log a 2 x 2 x log a c log a2 x2 x c 14. xii dx 1 x 2 15. xii dx x sin 1 x 2 dx cosd 16. xii dxx sin 1 x 2sin cosd dx cosd 1 sin 2 17. xii dxx sin 1 x2sin cosd dx cosd 1 sin 2 sin cosd cos 2 sin d 18. xii dxx sin 1 x2sin cosd dx cosd 1 sin 2 sin cosd cos 2 sin d cos c 19. xii dxx sin 1 x2sin cosd dx cosd 1 sin 2 sin cosd 1cos 2 x sin d cos c1 x2 20. xii dxx sin 1 x2sin cosd dx cosd 1 sin 2 sin cosd 1cos 2 x sin d cos c1 x2 1 x2 c 21. iii x 2 3dx 22. iii x 2 3dx x 3 tan dx 3 sec 2 d 23. iii x 2 3dxx 3 tan 3 sec 3 sec 2 d dx 3 sec 2 d 3 sec3 d 24. iii x 2 3dxx 3 tan 3 sec 3 sec 2 d dx 3 sec 2 d 3 sec3 d 3 sec sec 2 d 25. iii x 2 3dxx 3 tan 3 sec 3 sec 2 d dx 3 sec 2 d 3 sec3 du sec v tan 3 sec sec 2 d du sec tan d dv sec 2 d 26. iii x 2 3dxx 3 tan 3 sec 3 sec 2 d dx 3 sec 2 d 3 sec3 du sec v tan 3 sec sec 2 d du sec tan d dv sec 2 d 3sec tan 3 sec tan 2 d 27. iii x 2 3dxx 3 tan 3 sec 3 sec 2 d dx 3 sec 2 d 3 sec3 du sec v tan 3 sec sec 2 d du sec tan d dv sec 2 d 3sec tan 3 sec tan 2 d 3sec tan 3 sec3 d 3 sec 28. iii x 2 3dxx 3 tan 3 sec 3 sec 2 d dx 3 sec 2 d 3 sec3 du sec v tan 3 sec sec 2 d du sec tan d dv sec 2 d 3sec tan 3 sec tan 2 d 3sec tan 3 sec3 d 3 sec 3sec tan 3 sec3 d 3log sec tan 29. iii x 2 3dxx 3 tan 3 sec 3 sec 2 d dx 3 sec 2 d 3 sec3 du sec v tan 3 sec sec 2 d du sec tan d dv sec 2 d 3sec tan 3 sec tan 2 d 3sec tan 3 sec d 3 sec 3 x2 3x 3sec tan 3 sec3 d 3log sec tan 3 30. iii x 2 3dxx 3 tan 3 sec 3 sec 2 d dx 3 sec 2 d 3 sec3 du sec v tan 3 sec sec 2 d du sec tan d dv sec 2 d 3sec tan 3 sec tan 2 d 3sec tan 3 sec d 3 sec 3 x2 3x 3sec tan 3 sec3 d 3log sec tan 3x 3 x2 x 3 x 2 3 x 3dx 3log 2 3 3 3 3 31. x2 3 x 2 x 3dx x x 3 3log 22c 3 3 32. x2 3 x 2 x 3dx x x 3 3log 22c 3 3 x x2 3 3 x2 3 x x 2 3dx 2 log 2 3 c 3 33. x2 3 x 2 x 3dx x x 3 3log 2 2c 3 3 x x2 3 3 x2 3 x x 2 3dx 2 log 2 3 c 3 x 2 3dx x x2 3 32 log2 x2 3 x c 34. x2 3 x 2x 3dx x x 3 3log 2 2 c 3 3 x x2 3 3 x2 3 x x 2 3dx 2 log 2 3c3 x 2 3dx x x2 3 3 2 log 2 x2 3 x c Exercise 2E; 1, 2, 3, 5, 6, 7, 9, 13, 17, 19, 20