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    