Unit 2 Chapter 4 Answers



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  Page 1  of 46  Unit 2 Answers: Chapter 4 © Macmillan Publishers Limited 2013 Chapter 4 Integration Try these 41 (a) 5 2 5 2 1d 5  x x e x e c − − = + ∫  (b) 2 7 2 7 1d 7  x x e x e c − − = − + ∫  (c) π 1 cos 3 d sin 32 3 2  x x x c π   − = − +       ∫  (d) 1tan 5 d ln sec 52 5 2  x x x c  π π   + = + +         ∫  (e) 7 77  1 2 22 d 7 ln2 7ln2  x x x  x c c  = + = +   ∫  (f) 55 d ln5  x x  x c = + ∫   Try these 42 (a) 344 1d ln( 5)5 4  x x x c x = + ++ ∫  (b) 22 1d ln( 1)1 2  x x x c x = − +− ∫  (c) cosd ln(sin )sin  x x x c x = + ∫  (d) 22 2 3 1 1 6 2 1d d ln 3 2 13 2 1 2 3 2 1 2  x x x x x x c x x x x + += = + ++ + + + ∫ ∫   Try these 43 (a) 1222 d (1 ) d 1  x x x x x x − = ++ ∫ ∫   122 12 (1 ) d 2  x x x − = + ∫   122 1 (1 )122  xc − += +   2 1  x c = + +  (b) 87  cossin cos d 8  x x x x c = − + ∫    Page 2  of 46  Unit 2 Answers: Chapter 4 © Macmillan Publishers Limited 2013 (c) 2 32 3 2 1 1d (4 2)(2 2 3) d (2 2 3) 2  x x x x x x x x − += + + ++ + ∫ ∫   2 2 1 (2 2 3)2 2  x xc − + += +−   2 2 14(2 2 3) c x x = − ++ +   Try these 44 (a) 1 1 sin sin2 1d 1  x x e x e c x − − = +− ∫  (b) 1 1 tan tan2 1d 1  x x e x e c x − − = ++ ∫  (c) 3 3 1 2 1  1d 3  x x  x e x e c + + = + ∫  (d) cos cos sin d   x x  xe x e c = − + ∫   Exercise 4A 1   7 7 1d 7  x x e x e c = + ∫   2   4 2 4 2 1d 4  x x e x e c + + = + ∫   3   5 2 5 2 1d 2  x x e x e c − − −= + ∫   4   1 1d ln 4 54 5 4  x x c x = + ++ ∫   5   3 3d ln 7 27 2 7  x x c x = − +− ∫   6   2 2d ln 4 34 3 3  x x c x −= − +− ∫   7   1tan 2 ln sec 24 2 4  x x c π π   + = + +       ∫   8   2  1sec 3 d tan 32 3 2  x x x c π π   − = − − +       ∫   9   1 1d cos 2 d sin 24 2 4sec 24  x x x x c x π π   = − = − +   π     −   ∫ ∫   10   1d sin ( 2) d cos ( 2)cosec ( 2)  x x x x c x = + = − + ++ ∫ ∫   11   22 1 1d sec (3 1) d tan (3 1)cos (3 1) 3  x x x x c x = + = + ++ ∫ ∫    Page 3  of 46  Unit 2 Answers: Chapter 4 © Macmillan Publishers Limited 2013 12   3 3 32 2 6 d 2 3 d 2  x x x  x e x x e x e c = = + ∫ ∫   13   cos cos cos sin d sin d   x x x  x e x xe x e c = − − = − + ∫ ∫   14   1 1d 2 d 22  x x x e x e x e c x x = = + ∫ ∫   15   2 2 2 1 1d 2 d 2 2  x x x  xe x xe x e c − − − −= − − = + ∫ ∫   16   3 2 6 4 2 6 4 2 1 1 1( ) d 2 d 6 2 2  x x x x x x x x e e x e e e x e e e c − = − + = − + + ∫ ∫   17   22 2 1 2 1d d ln 99 2 9 2  x x x x x c x x = = + ++ + ∫ ∫   18   cos 1 2cos 1d d ln 2sin 12sin 1 2 2sin 1 2  x x x x x c x x = = + ++ + ∫ ∫   19   2 4secd 2ln 2tan 52tan 5  x x x c x = − +− ∫   20   233 2 2d ln 55 3  x x x c x −= − +− ∫   21   333 1d ln 11 3  x x x e x e ce = + ++ ∫   22   [ ] 22 arcsin 1d arcsin21  x x x c x = +− ∫   23   2 2  1tan 3 1 d sec 3 d sec (3 ) d ln sec(3 ) tan(3 )3  x x x x x x x x c + = = = + + ∫ ∫ ∫   24   2 1 1 1sin d cos  x c x x x    = +       ∫   25   54  cossin cos d 5  x x x x c −= + ∫   26   sin4 sin4 1cos4 d 4  x x e x x e c = + ∫   27   3 3 32 2 1 1d 3 d 3 3 t t t  t e t t e t e c = = + ∫ ∫   28   112200 1 1 1 1 10d ln 9 ln10 ln9 ln9 2 2 2 2 9  x x x x    = + = − =   +     ∫   Try these 45 (a) 2 10 d   x  xe x ∫   2 Let  u x =   d 2 d  u x x =   1d d 2 u x x ∴ =   When 0, 0  x u = =  







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