2022 Chp 6A Techniques of Integration APQ (RVHS)

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6A Integration Techniques (Solutions to Tutorial and Assignment) 28 Additional Practice Questions (H2 Chap 6A Integration Techniques) 1. [2008/Promo/AJC/Q8(b)] (i) Given that   sinsin xy x e , find d d y x . Hence, or otherwise, show that sin sin sin(sin 2 ) d 2(sin ) 2x x xx e x x e e C   , where C is an arbitrary constant. [3] (ii) Find sin(sin )(sin 2 ) d xx x e x . [3] Solution:

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6A Integration Techniques (Solutions to Tutorial and Assignment) 29 2. [2008/Promo/ACJC/Q4] Express   231 146 22 23   xx xxx in the form 231 22    x DCx x BAx where A, B, C and D are constants to be determined. [3] Hence, or otherwise, find    3 2 2 2 6 4 1 1 3 2 x x x dx x x      . [3] Solution:

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6A Integration Techniques (Solutions to Tutorial and Assignment) 30 3. [2008/Promo/CJC/Q14] (a) (i) Express   2 2 41 73 xx xx   in partial fractions. [4] (ii) Hence find   xx df , where   2 2 41 73)f( xx xxx   . [2] (b) By substituting 53 2  xu , find the integral xxx d 533 2   . [3] Solution:

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6A Integration Techniques (Solutions to Tutorial and Assignment) 31 4. [2008/Promo/HCI/Q2] Solve  cos ln dx x x . [5] Solution:

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6A Integration Techniques (Solutions to Tutorial and Assignment) 32 5. [2008/Promo/NJC/Q11(a)] By considering 1 ( 1 2 )x A x B     , where A and B are constants, or otherwise, find 2 1 d 3 x x x x    . [4] Solution:

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6A Integration Techniques (Solutions to Tutorial and Assignment) 33 6. [2008/Promo/SAJC/Q10] Find the following integrals : (a) 2 4 9 4 dx x [3] (b)   sin 2cos 2 cos 2 xx e x dx [3] (c) 2 2 1 4 5 x dxx x    [4] Solution:

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6A Integration Techniques (Solutions to Tutorial and Assignment) 34 7. [2008/Promo/TJC/Q11] Find (a) 2cot 3 dx x . [3] (b) Find   1 2sin dx x x  . [4] (c) Given that  3 2 2 4x A x B    for all values of x, find the constants A and B. Hence or otherwise, find 2 3 2 d 4 6 x x x x    . [4] Solution:

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6A Integration Techniques (Solutions to Tutorial and Assignment) 35 8. [2010/Promo/HCI/Q12] (a) Write down the constants A and B such that, for all values of x ,  2 5 1x A x B    . Hence find 2 2 5 d2 5 x xx x    . [4] (b) Find the derivative of   2tan x . Hence find   3 2 2sec dx x x . [4] (c) By using the substitution ux 1 , find the exact value of 4 22 2 1 d 4 x x x  . [5] Solution: 8(a) 2, 7A B    2 2 2 2 2 2 2 1 2 5 2 2 7d d2 5 2 5 2 2 7 d d 2 5 2 5 1ln 2 5 7 d 1 4 7 1ln 2 5 tan + 2 2 x x x xx x x x x x x x x x x x x x x xx x C                                  (b)     2 2 2d tan 2 secd x x xx          3 2 2 2 2 2 2 2 2 2 sec d sec d ' sec 1' 2 tan 2 x x x x x x x u x v x x u x v x                         2 3 2 2 2 2 22 2 2 2 2 2 sec d tan tan d 2 2 sin1 1= tan d = tan + ln cos + 2 2 2 2 cos xx x x x x x x x xx x x x x x C x        (c) 2 1 1 d dx x uu u   1 12 2 and 4 42 2 x u x u      14 2 2 12 22 2 4 1 1 d d 4 1 4 x u x x u        1 1 2 2 1 4 1 sin 22 1 2 4 6 24 u             

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6A Integration Techniques (Solutions to Tutorial and Assignment) 36 9. [2010/Promo/RVHS/Q11] (a) Using the substitution exu , find 1 de 2ex x x . [4] (b) By expressing 4x – 5 in the form A(2 – 2x) + B, show that 1 20 4 5 3 π d = 63 2 x a b x x x      , where a and b are constants to be found. [6] Solution: a. x x duu e e u dx    . Then:   2 2 2 -1 1 1 1 22 1 2 1 2 1 tan 2 2 2 tan 2 2                             x x x dudxe e u u u duu du u u c e c b.   1 1 2 20 0 1 1 2 20 0 1 1 2 20 0 11 2 -1 0 0 1 -1 0 4 5 2(2 2 ) 1 3 2 3 2 2 2 1 2 3 2 3 2 2 2 1 2 3 2 4 1 1 2 2 3 2 sin 2 1 1 2 2 4 2 3 sin 2 x x dx dx x x x x x dx dx x x x x x dx dx x x x xx x                                                         1 0 1sin 2 4 3 8 0 6 24 3 48 6                   

H2 Mathematics (9758) JC1 River Valley High School Mathematics Department, 2022 Ch 6A Integration Techniques (Solutions to Tutorial and Assignment) 37 10 2014 VJC/ Promo/ 4 (a) Find   2tan 3 dx x . [2] (b) Find 2 2 3 d2 5 x xx x    . [4] (c) Differentiate 1 2sin ( )x with respect to x. [1] Hence find the exact value of   1 43 4 1 2 41 2 d sin 1 x x x x        , simplifying your answer. [4] Solution: 10(a) (b) (c)       2 2tan 3 d sec 3 1 d 1 tan 33 x x x x x x C            2 2 2 3 d2 5 2 2 5 d2 5 x xx x x xx x            22 2 2 1 d 5 d2 5 1 4 x x xx x x        2 1 1 1ln 2 5 5 tan 2 2 xx x C                 2 1 2 5 1ln 2 5 tan 2 2 1 4 0 for all real values of xx x C x x                 1 2 4 d 2sind 1 xxx x          1 1 4 4 1 4 3 3 44 4 1 21 2 41 1 2 2 3 1 2 4 1 2 2 1 1 d d 2 sinsin 1 1 ln sin2 x x xx x xx x x                           1 11 3 1 1ln sin ln sin ln ln2 2 2 2 3 6 1 ln 22                               