12 Integration TutSol

Page 1 of 18 Integration Techniques Section 1: Discussion Questions (Students are to attempt all these questions.) 1 Integrands in the form    f ' f n xx (a) 2 3 1 d 3 x x xx     1 3 22 33 x x c 2 3 1 d 3 x x xx       1 2 1 2 32 3 13 2 31 3( 1) 1 2 d 33 3 3 3 xxx x c x x c xx        (b)    523 1 3 dx x x x   631 318 x x c    523 1 3 dx x x x      562 3 311 3 1 3 d 33 18 x x x x x x c      (c) 5sin cos dx x x 6cos 6 x c   6 55 cossin cos d sin cos d 6 xx x x x x x c    (d) 5sec tan dx x x 5sec 5 x c   5 54 secsec tan d sec tan sec d 5 xx x x x x x x c   (e)     3cos 3 sin 3 dx x x   4sin 3 12 x c     3cos 3 sin 3 dx x x         4 34 sin 31 1 13cos 3 sin 3 d sin 33 3 4 12 xx x x c x c     2 Integrands in the form     f' f x x (a) 1 d (1 ) x xx  2ln 1 xc  

Page 2 of 18 Integration Techniques 1 d (1 ) x xx  112 d 2ln 1 2 (1 ) x x c xx      (b) d3 x xx 3ln 3x x c     xx x d3 3 3 3d 1 d 3ln 333 x x x x x cxx          (c) cot d ln sin c  cot d cos d ln sinsin c   

Page 3 of 18 Integration Techniques 3 Integrands requiring use of trigonometric identities (a) 2sin 3 dxx 11 sin 626 x x c  2sin 3 dxx  1 1 sin 61 cos6 d2 2 6 xx x x c       (b) sin cos d22 xx x 1 cos2 xc sin cos d22 xx x 11 sin d cos22 x x x c   (c) 1 d1 cos 2 xx 1 cot2 xc 1 d1 cos 2 xx 2 2 1 1 1 d cos d cot2sin 2 2 x ec x x x cx    (d) 1 d1 cos xx 1tan 2 xc   xx dcos1 1 2 22 2 11 d sec d tan2cos 2 2 x x xx x c    (e) 53sin cos d22 xx x 11cos 4 cos82 x x c   53sin cos d22 xx x  1 1 cos 4 1(sin 4 sin )d cos cos 4 4cos2 2 4 8 xx x x x c x x c           (f) 3cos dxx 3sinsin 3 xxc 3cos dxx 3 22 sincos (1 sin )d cos cos sin )d sin 3 xx x x x x x x x c       OR 2 2 2 23 1cos (1 sin )d cos (cos 2 sin )d 2cos cos 2 d cos sin d2 1 1 1 1(cos3 cos )d cos sin d sin 3 sin sin2 6 2 2 x x x x x x x x x x x x x x x x x x x x x x c                   (g) 2 4 3cos cos d22 a a aa                1 124a 

Page 4 of 18 Integration Techniques       2 22 44 4 3 1 1 sin 2 sincos cos d cos 2 cos d2 2 2 2 2 1 1 1 10 1 ( ) 1 22 2 4 2 a aa aa a a a a a aa aa aa                                       (h) 1 0 sin( 1) sin( 1) d     sin 2 4         11 00 1 0 1sin 1 sin 1 d cos 2 cos 2 d 2 1 sin 2 sin 2 1 sin 2 sin 2 002 2 2 2 2 4                               4 Integrands requiring long division and/or partial fractions (a) 2 1 d2 xxx 11ln ln 222 x x c   2 1 d2 xxx 1 d( 2) xxx  1 1 1 d22 xxx