8.Integrations Solutions 2023 (RVHS)

River Valley High School, Mathematics Department 2023 Integration T1 Solutions (Techniques of Integrations) 1   2 1 2tan dxxe e x       2 2 1 2 2 22 1 1 2tan d22 1 x x x x x ee e e x e        2 1 2 4 11 tan d2 1 xx xe e x e       4 2 1 2 4 1 tan d21 x xx x ee e x e       2 1 2 411 tan ln 124 x x xe e e C    21 4 2 x dx xx 22 1 4 4 1 4 1 4 2 1 4 2 x dx dx x x x x         2 2 11 1 4 22 321 2 x x dx x        21 2111 1 4 2 sin2 23 xx x C       2 (i)    xdxxx dxxx coscos sin cxxx  cossin uduudx cossin2 (ii)   cx x x cuu udu duu uduu uu                11ln2 tansecln2 sec2 cos 12 cossin2 cos1cos 1 22

River Valley High School, Mathematics Department 2023 Integration T2 3 (a) 0 6 5sin 3cos d cos sin xx xxx     0 6 cos sin 4(cos sin ) d cos sin x x x x xxx     0 6 cos sin 4 dcos sin 0 ln cos sin 4 6 2 3 1 ln32 xx xxx x x x              (b)         1 1 1 2 2 22 2 2d 1 1 2 1d2 xx x x x xx          22 2 1 1 xx x    2 2 12 1 x x     12 2 2 2 1 22 2 22 0 1 22 21 2 0 12 d 1 1 13 2 2 d 1 2 11 0 132 d 1 2sin 2 1 0 3 34 x x x x C x x x x x xx x x x x x x                     4    x xx x d 22lnln 2lnln tx e2    tt x tt t dd d 22lne2lne2 2lne2ln t t x e2d d     t t t t t de2 22ln2lne2 2ln2ln    t t t d 2

River Valley High School, Mathematics Department 2023 Integration T3  22 d 2 t t t     1 2 22d 2 tt t     (shown)    4 2 e2 e2 d 22lnln 2lnln x xx x     4 2 2 1 d 2 22 t t t     4 2 2 1 2 3 2423 2      tt          2 1 2 3 244243 2    2 1 2 3 2423 2           423 22 2 1 3 216 Therefore 16a , 3b 5 . 6 (a) Let 2 22 2 5 13 (2 2) 2 5 2 5 x x B x C Ax x x x         222 5 13 ( 2 2 ) (5 2 )x x Ax A B x A B C         Comparing coeffs: 2 12 2 5 2 5 2 13 2 A A B B A B C C             cos cosd e e sind xx xx  cose sin 2 dx xx   cose 2sin cos dx x x x    cose sin 2cos dx x x x   cos cos2e cos 2 e sin dxx x x x   cos cos2e cos 2exx xC  

River Valley High School, Mathematics Department 2023 Integration T4   2 2 2 2 2 2 5 13 1 2 2 2d 2d d d2 5 2 2 5 1 2 x x x x x x xx x x x x               21112 ln( 2 5) tan22 xx x x C        (b) 2 1 1 ln d e nx x x  = 2 1 1ln d ennxxxx n n x      = 2 2 1 1ln en nxxx nn     = 22 (2 ) 1 1ln(2 ) (2 ) 0 n neee n n n       =  2 1 (2 ) ln 2 1 (2 ) 1nnn e en     7             1 2 21 1 1 12 22 2 1 2 1 11 2 1 122 1 2 122 11 2 4 2 2 1 11 1 1 1 1 2 2 2 2 1 412 x x xx xx xxxx e dx e e dx e dx ee e e e e e e e e                                     8 12cos dx x x  22 12 4 23 12 4 2 1 2 4 2cos d22 1 14cos d24 1 1cos 122 x x x xx x xx xx x x x x C                9      2 2 ln 2 d 25 2 ln 2 x x xx    =   2 2 21 ed2e 25 2 u u u u u   1 e 2 e2 uuxx   2d e d uxu

River Valley High School, Mathematics Department 2023 Integration T5 = 2 2 d 25 2 u u u = 2 2 1 2 25 25 d2 25 2 u u u     = 2 1 251d2 25 2 u u   =    1 1 5 225 ln2 2 2 5 5 2 uuc u       = 1 5 5 2 ln2 2 2 5 2 uuc u      =      5 2 ln 215 ln ln 22 2 2 5 2 ln 2 x xc x     10 (a) 0 0 d d22 a a aax x k x x     22 1 3 1 1( )( ) 2 ( )2 2 2 2 2 2 1 4 4 aaa a a k a a ka k       (b) (i) 2d ( sin 2 )d xxx = 22 sin 2 2 cos 2x x x x  (ii) 22(2 cos 2 2 sin 2 )d sin 2x x x x x x x x a - a y

River Valley High School, Mathematics Department 2023 Integration T6 22 2 2 2 (2 cos 2 ) d 2 sin 2 d sin 2 [ cos 2 cos 2 d ] sin 2 1[ cos 2 sin 2 ] sin 22 1cos 2 sin 2 sin 22 x x x x x x x x x x x x x x x x x x x C x x x x x C                   22 1 1 1(2 cos 2 ) d cos 2 sin 2 sin 22 4 2x x x x x x x x C    11(a) (i)       2 2 2 2 sin1tan d d 2 cos xx x x x x x     21 ln cos2 xc  (ii) 22 1 2 1 1dd3 2 3 xx xxx x x x       22 1 11 24 1 2 1 1 1 dd2 3 2 x xxxx x     21 2111ln 3 tan2 11 11 xx x c       (b) (i)   1 122 0 sin dx x x    1 123 212 2 4 0 0 sin d2 1 xx xx x      1 123 212 2 4 0 0 14sin d24 1 xx xx x       1 2 21 2 4 0 1sin 122 x xx   31 24 4 2    (ii) Since 0 < b < 1 , 1 0 x x b dx = 1 0 ( ) ( ) b b x x b dx x x b dx    13 2 3 2 0 3 2 3 2 b b x bx x bx             3 1 3 3 2 bb  

River Valley High School, Mathematics Department 2023 Integration T7 12   12 1 2 2 12 13 1 12 2 2 331 (tan 2 ) 14 1 (tan 2 ) 12 (tan 2 )26 14 1 tan 264 n nn x dx x x dx x x n               As n → ∞, 1tan 2 2n   . 331 3 1 2 2 (tan 2 ) 1 7 6 2 4 38414 x dx x                    13 (i)   2d 2d x x = 212 ln 2x (ii)    2 21 22 22 21 2 ln 2 d 1 2 ln 2 d2 1 2 2 d2 11 222 2ln 2 12 2ln 2 xx x xx xx x x xx xx xC xC               14 (a) 22 2 21(ln ) d [ (ln ) ] [2(ln )( )( )]d2 2 2 2 [ (ln ) ] 2 (ln )d22 x x xx x x x x xxxx     2 21[ (ln ) ] 2[ (ln ) ( )( ) d ]2 2 2 xxx x x x x    2(ln ) 2 (ln ) 222 xxx x x c    (b) x ax 4 3  0 44  x ax 0 2222  x )ax)(ax(

River Valley High School, Mathematics Department 2023 Integration T8   0 22  x )ax(ax)(ax Since 022  ax ,     0x a x a x   0 xa or xa For 1 xa , 43 0ax x For 3ax , 43 0ax x 43 3 1 d axx x = 44 3 33 1 ( )d ( )d a a aax x x xxx    44 4 4 3 1[ ln ] [ ln ]44 a a xx a x a x    44 4 4 4 4 44 24 4 1 81[ ln ( )] [ ln 3 ( ln )]4 4 4 4 412 ln ln 3 22 41ln 3 2 2 aa a a a a a aa a a aaa               15 00 dd cc x c x c x x     02 2 c xcx   2 22 ()() 2 1 2 ccc cc      23 2 c 22 00 d d d c c c c x c x c x x x c x       222 022 cc c xxcx cx             ‒ a − − + a 0 +

River Valley High School, Mathematics Department 2023 Integration T9 2 2 2 2 2 2 2 4 22 2 2 c c cc c c c                 02 22 0 3dd 2 c c x c x k x c x c kc       3 2k Alternative: 02 0 dd c c x c x k x c x     Area A1 = k (Area A2 + Area A3)   22 1 1 1(2 ) ( ) ( )2 2 2 1 32 3 2 c c c k c c c c c kc k       16 (a)    2 d12 x xxx 141d 12 ln 1 4ln 2 xxx x x x c            (b)(i)   12 4 d2sind 1 xxx x    x y c c y x c 2c –c 2c A1 A2 A3

River Valley High School, Mathematics Department 2023 Integration T10 (b)(ii)         12 0 22 12 40 0 2 1 2 4 0 2 1 2 4 sin d 2sin d22 1 1sin 2 124 1 1 1sin 12 2 2 4 2 n n n n x x x x x x xx x x xx n nn                      From