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 = =