9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =

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1 A II f() f() F () f() F () = f() C (F () + C) = F () = f() F () + C f() F () G() f() G () = F () 39 G() = F () + C C f() F () f() F () + C C f() f() d f() f() C f() f() F () = f() f() f() d = F () + C C 89

2 9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 = 3 = d d f() f() 5. d () 5 d () 3 (3) 3 d (4) 3 d (5) d (6) d

3 5.. 9 B II kf() d = k f() d k {f() + g()}d = f() d + g() d 3 {f() g()}d = f() d g() d 5. ( )( ) 3 + d = = d 3 d = d + = 3 log + C ( 3 + ) d d C d d ( )( 3) () d () d 3 4

4 9 5 (3) + d (4) ( ) d (5) d (6) ( 3t t ) dt

5 C (sin ) = cos (cos ) = sin (tn ) = cos (e ) = e ( ) = log sin d = cos + C, cos d = sin + C 5.3 () d cos = tn + C, e d = e + C, ( sin + 3 cos )d = d = sin d + 3 log + C cos d 5.4 () = cos + 3 sin + C ( ) tn d = cos d + tn = cos (3e )d = 3 = tn + C e d d = 3e log + C 5.3 cos 3 () (cos sin )d () d cos

6 94 5 (3) d sin (4) ( tn θ) cos θ dθ (5) 4 d (6) (3 e )d 5.. A f( + b) f() F () f( + b) F () = f() d d F ( + b) = F ( + b) = f( + b) 93 3.() f( + b) F () = f() f( + b)d = F ( + b) + C

7 () () (3) 3 + d = (3 + ) + ) d = 3 (3 + + C + = 9 (3 + ) 3 + C = 9 (3 + ) C cos 4 d = sin 4 + C 4 e d = e + C 5.4 () (3 + ) 4 d () (4 3) 3 d (3) + d (4) d (5) sin d (6) e 3 d

8 96 5 B F () f() t = g(t) = F () t g(t) d d = f() d dt = d d d dt = f()g (t) = f(g(t))g (t) () f() d = f(g(t))g (t) dt = g(t) g (t) = d dt f() d = f(g(t)) d dt dt 5. d = t = t d dt = t d = ( t )t( t)dt = (t 4 t )dt ( ) t 5 = 5 t3 + C = 3 5 t3 (3t 5) + C = 5 (3 + )( ) + C = t

9 5.6 () ( ) 4 d () d (3) + d

10 98 5 C f(g())g () () t u () f(g())g () d = f(u) du u = g() g () = du d g () d = du 5. () + d () cos sin d () ( + ) = + = u + d = + d = + ( + ) d = u du = 3 u 3 + C = 3 ( + ) + + C () (cos ) = sin cos = u cos sin d = cos ( sin )d = cos (cos ) d = u du = 3 u3 + C = 3 cos3 + C

11 5.7 () 3 + d () sin 3 cos d (3) log d (4) e d

12 5 98 f(u) = u g () g() d = du = log u + C = log g() + C u g () 3 d = log g() + C g() 5.3 () d () tn d 3 ( () 3 d = 3) 3 d = log 3 + C sin (cos ) () tn d = cos d = d = log cos + C cos () + d () e e + d (3) d tn

13 5.. D {f()g()} = f ()g() + f()g () f()g() f()g() = f ()g() d + f()g () d 4 f()g () d = f()g() f ()g() d 5.4 cos d cos d = (sin ) d = sin () sin d = sin sin d Z g () d = sin + cos + C g () d = g() g() d

14 5 5.9 () sin d () e d 5. log d log = (log ) = (log ) () log d = (log ) () d = (log ) (log ) d = log d = log + C

15 5. () log d () log d (3) log d

16 A d () d () + + () + = + 3 () + + d = + ( + 3 ) d + = + 3 log + + C = ( ) + d = ( ) d + = (log log + ) + C = log + + C () 5. ( + )( + ) = + + b + b d ( + )( + )

17 5. () + d () 4 d (3) 3 + d

18 6 5 B sin cos α α = cos + cos α sin α α = sin α cos α = sin α cos β = {sin(α + β) + sin(α β)} cos α cos β = {cos(α + β) + cos(α β)} sin α sin β = {cos(α + β) cos(α β)} 5.6 () sin d () sin 3 cos d () () cos sin d = d = ( cos )d = ( ) sin + C = sin + C 4 sin 3 cos d = (sin 4 + sin )d = ( 4 cos 4 ) cos + C = 8 cos 4 cos + C 4

19 5.3 () cos d () sin 3 d (3) cos cos 3 d (4) sin 3 sin d

20 d () sin = tn + C () d tn = tn + C () d () d 4 e +

21 5.. 9 (3) log( + ) d (4) (sin + cos ) d (5) sin 3 d (6) cos 4 d

22 5 d 3 + cos () cos () + cos = cos

23 4 F () F () = F () = [ ( () ) () tn tn = ] sin () () 4 + C () log(e log( + ) + ) + C (3) + log + + C (4) cos +C (5) cos + 3 cos3 +C (6) 3 sin 4+ 4 sin C 3 () tn + sin + C 4 F () = log ( + ) + () tn + C C = log

24 A b II f() f() d b [, b] f() F () b f() d = [ ] b F () = F (b) F () b f() d f() b [, b] f() b f() d = f() S = = b S b f() d =f() b 5.6 () () 4 π d = [ 3 [ sin θ dθ = cos θ ] 4 ] π = 3 (4 4 ) = 4 3 = ( ) =

25 () d () 8 3 d (3) π cos θ dθ (4) e d (5) d (6) d

26 4 5 B II b b b b b 5.7 () k f() d = k b {f() + g()}d = {f() g()}d = f() d = f() d b b b f() d = f() d f() d = c f() d + f() d + f() d b c k b b f() d g() d g() d 3 4 ( 3 d = 4 ) d d = 3 4 d [ ] [ = 3 log 4 ] = 3(log log ) 4 ( ) + = 3 log () π π sin d = = cos d = [ sin ] π = π π ( cos )d

27 () + d () ( + ) 3 d (3) (e t e t )dt (4) π sin d (5) π cos d (6) π sin 4θ cos θ dθ

28 6 5 C f() c f() c b f() f() b b f() d = c f() d + b c { f()} d = f() = = b c = f() b = f() π 5.7 sin d π sin = sin = sin π π sin = sin π π π sin d = = π sin d + [ cos ] π + π π [ cos = ( + ) + ( + ) = 4 ( sin )d ] π π

29 () π cos d = cos π π () e d = e

30 A F () f() b g(t) = g(t) d dt F (g(t)) = f(g(t))g (t) = g(α) b = g(β) α = g(t) β t β α f(g(t))g (t) dt = [ ] β F (g(t)) = F (g(β)) F (g(α)) α = F (b) F () = b f() d = g(t) = g(α) b = g(β) b f()d = f(g(t))g (t)dt t b α β 5.8 ( ) 4 d = t = t d dt = t t = t t ( ) 4 d = = ( t)t 4 ( )dt = [ 5 t5 6 t6 ] = 5 6 = 7 3 (t 4 t 5 )dt

31 () ( ) 5 d () 5 d

32 5 5.8 > d = sin θ d dθ = cos θ θ cos θ θ π > = ( sin θ) = cos θ = cos θ d = π = π = π 4 ( cos θ) cos θ dθ = π cos θ dθ [ + cos θ dθ = θ + ] π sin θ 5.8 = = d

33 () d () 3 4 d (3) 3 d 4

34 5 d 5. + = tn θ tn θ + = cos θ = tn θ d dθ = cos θ θ θ π 4 d + = = π 4 π 4 tn θ + cos θ dθ [ ] π 4 dθ = θ = π 4 + = tn θ 5.9 () 3 d +

35 5.. 3 () d + 4 B f() f( ) = f() f( ) = f() cos sin tn 4 + cos

36 4 5 f() f() f() d = ` f() f() d = ` f() d f() d = f() d + f() d f() d = t d dt = t f() d = f( t) ( ) dt t = = f( t) dt f( ) d f() d = {f( ) + f()}d f() f( ) = f() f( ) = f() 5.9 () f() = cos π π [ ] π cos d = cos d = sin = π () f() = sin π π sin d =

37 5. () ( ) d () (e e ) d (3) 4 d (4) π π sin d C b [ ] b b f()g () d = f()g() f ()g() d π 5.9 cos d π π cos d = = [ sin = π π (sin ) d ] π π sin d = π [ cos ] π () sin d = π

38 () π sin d () e d (3) log d

39 ( + ) 3 ( ) d 5..3 A f(t) dt F (t) = f(t) f(t) dt = F () F () d d 5.4 () sin t dt () f(t) dt = f() t log t dt

40 G() = ( t) cos t dt t G() = cos t dt = cos t dt cos t dt G () = () = cos t dt ( d cos t dt + d t cos t dt cos t dt + cos cos = ) cos t dt d t cos t dt d [ ] sin t = sin 5.5 G() G () G () G() = ( t)e t dt

41 5.. 9 B f() = = f() = S S = S = [ 3 d = 3 ] = 3 = [, ] n n S n n S n S S S n { ( S n = ) ( ) + + n n n = n ( ) k = n k n n n 3 k= k= = n(n + )(n + ) n 3 6 lim S n = lim n n 6 ( ) } 3 ( n ) + + n n ( + ) ( + ) = n n 3 n n n n lim S n = S n

42 S n 3 3 T n lim T n = S n = T n n n n n

43 5.. 3 f() [, b] f() S = f() d = f() [, b] n b,, 3,, n b =, b = n, n = n b S n n S n S n = f( ) + f( ) + f( 3 ) + + f( n ) n S n S f() n b lim f( k ) = f() d = b n! k= n k = + k = b = lim n n n f k= ( ) k = n f() d = n, k = k n

44 3 5 ( 5.4 lim n n + + n + + n ) n + n n ( ) k f n n n k= S ( S = lim n n + n + + n ( ) n = lim n n k= + n k n ) n f() = + S = lim n n n f k= ( k n ) = f() d = 5.7 () lim n n 5 ( n 4 ) [ ] d + = log( + ) = log ( ) 3 n () lim n n + 3 n + 3 n n 3

45 C f() [, b] b = f() = = b f() f() = b f() d > [, b] f() g() f() g() b f() d b f() d g() d =f() b f() = g() 5. () + + ( + ) () d + + > () + + ( + + ) ( + ) + + ( + ) () () + + = ( + ) d + + > d ( + ) = d + + > [ + d ( + ) ] =

46 () () d + + < log

47 f() = log n > n n k k < < k + > k + k+ k+ k d > d k k + k k < < k + > k + = k+ k+ k d > k k + d k+ d k > k + k + k k + k =,, 3,, n d 3 + d 4 + d n d n > n n = [ ] n d = log = log n log = log n > n

48 f() = n > log(n + ) n k = k k +

49 () 4 + d () π 3 tn θ dθ (3) π π 6 cos θ dθ (4) (3 ) d

50 38 5 (5) e d (6) 4 log d 6 () d () + d

51 b 7 f( + b ) d = b f() d () > () sin θ dθ () log t dt 5 () () log (3) π + 6 (6) 3 3 log 8 [ [ (6) 9 3 log (4) log 3 (5) ] 4 4 ( 6 () π4 () π ) 8 4 [ b 7 + b = t f( + b )d = 8 () sin sin () (4 ) log ( ) e 3 ] d b ] f(t)( )dt

52 A = f() = f() [, b] f() = f() = = b S =f() S = b f() d S b 5. = e = =e = S S = e d = [ ] e = e e S 5.3 () = 3 = = = 3

53 () = + = = 3 = + 3 [, b] f() S S = b { f()} d = f() S b S = f() 5. S = ( + ) ( + ) = S =, =( + ) S S = { ( + ) } d = [ 4 = ] = 4 3 ( ) d

54 () = ( ) = ( ) () = cos ( π 3 ) π = cos π 3 π 5.3 = ( + )( ) = ( + )( )

55 II b f() g() = f() = g() =f() = = b S S b S = {f() g()} d b =g() 5. = sin, = cos = sin S π 5 4 π S π π 4 = cos = π 4, 5 4 π sin cos S = 5 4 π π 4 (sin cos ) d = [ ] 5 4 π cos sin = π () = = = =

56 44 5 () + 4 = 5 = = + 4 = 5 4 = g() c d g() = g() = c = d S S = d c g() d 5.3 = log = S S = log = e e > [ ] S = e d = e = e = log 5.34 = = +

57 B 5.6 > b > + b = πb S b = b b S = b d = b d d S = b π = πb =

58 46 5 θ = (θ sin θ), = ( cos θ) π π (θ =) (θ =π) (θ =π) 5.7 > = (θ sin θ), = ( cos θ) ( θ π) S =, π S = π d S = π d = (θ sin θ) S π d = ( cos θ) dθ π θ θ π S = = π π = π d = π d dθ dθ ( cos θ) ( cos θ) dθ π = ( cos θ + cos θ) dθ ( cos θ + + cos θ ) dθ [ 3 = θ sin θ + ] π 4 sin θ = 3π

59 () = 3 cos θ = sin θ ( θ π) ( ) θ = π 3 (θ = π) 3 (θ = ) () = cos θ = cos θ ( θ π) (θ = π) (θ = ) ( ( ) θ = π θ = 3 π) 4 4 ( ) θ = π

60 A 3 α α β V α β S() A B A P B b < b b P γ S() V S() V γ β V = b S() d < b b n S( k ),, 3,, n =, b = n, b n = A B k k n b n V n = S( ) + S( ) + + S( n ) = n S( k ) n V n V 3 n b V = lim S( k ) = S() d n k= k=

61 r h V V = 3 πr h S() S h r S() S S = πr : h S() : S = : h S() = S h : b : b V = h S() d = h = S h h3 3 = 3 Sh = 3 πr h S h d = S [ ] 3 h h S h V V = 3 Sh S() S h

62 AB P AB QR h P A B V AB P QR = PR = QR A S() R S() h B P Q S() = h = h V = S() d = h d = h π = π h 5.38 AB 45 A 45 B S()

63 B = f() = = b V =f() b (, ) f() =f() f() S() S() = π f() = π{f()} S() b V = π b {f()} d = π b d < b 5.6 r V V = 4 3 πr3 r = r V = π = π r r r r r d (r ) d = π (r ) d [ ] r = π r 3 = πr3 r r r = r r

64 () = = () = sin ( π) = sin π 5.4 > b > + b = b b

65 < r < + ( ) = r V V + ( ) = r = ± r =+ r = + r = r = r V = = r r = r = r V r r V = π r r ( + r ) d, V = π r V = V V = 4π r d r = 4π πr = π r r r ( r ) d 5.8

66 = 4 = =4 = = e = e = e e = e

67 C =g() = g() = c = d V c d g() V = π d {g()} d = π d c c d c < d 5.7 = = 3 V V = π = π 3 3 d ( + ) d [ = π + = 5 π ] 3 3 =3 =

68 () = 4 = 4 =4 = () = =

69 π = sin = sin = sin π 3 π = sin + = + = = 3 4 (, 3) = 3 4 3

70 58 5 π 3 = sin = sin = sin π 3 π = sin 9 5 [ π 3 (sin sin ) d π [ ( ] ) d [ ( ) d [ π π 3 ] π π 3 ] (sin sin ) d π ] sin 3 d π sin d

71 A () e e + () ( ) 3 (3) 3 +

72 6 5 (4) cos 5 (5) e e (6) log( + )

73 () π sin 5 cos d () e e + d (3) e log d (4) 4 ( 4)( ) 3 d

74 6 5 (5) d (6) 3 t ( + t ) dt π 3 m n sin mt sin nt dt m n m = n

75 4 lim n n ( n + + n ) n + 3 n = ( ) = ( ) 6 = e (, e) S V =e =e

76 B 7 d () ( + 3) () d sin 8 I = (e ) d I

77 π 9 f() f(sin ) d = π f(cos ) d π f() = + f(t) sin t dt f() > f() f(t) dt = log f()

78 66 5 = log e 3 = (θ sin θ) = ( cos θ) ( θ π) > π

79 7 () ( + 3) = + b c + 3 () sin = sin sin = π sin cos 9 f(t) sin t dt = () 3 (e + ) e + + C () ( ) + C (3) log + C ( ) (4) sin 3 sin3 + 5 sin5 + C (5) log e e + + C (6) ( + ) log( + ) + C () () log(e + ) log (3) e3 + (4) 3 (5) π (6) π [ 4 (4) ( 4)( ) 3 d ( 4)( ) 3 d (5) = sin θ ] (6) t = tn θ 3 m n m = n π 4 ( [ ] d ) [ S = 4 ] 3 d 6 S = e [ V = e 3 ] π S = e d e d 6 7 () 9 log C () cos log + cos + C [ 8 = 3 (e 7) I = 3 + ] (e ) [ π ( ( π )) ] 9 = f cos π d = t [ π π ] f() = π f(t) sin t dt = (t + ) sin t dt =

80 68 5 f() = [ ( > ) = f( ) = f( ) = = ] log = e 4 3 5π 3 [ V = π [ ( e ), e π e π d = π ( cos θ) ( cos θ) dθ ] log d π π = π 3 ( cos θ) 3 dθ = π 3 ( 3 cos θ + 3 cos θ cos 3 θ) dθ π cos 3 θ dθ = ]

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