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/Q