TMJC H2 Chapter 10 Integration Techniques Discussion Questions_Solutions 2024

Chapter 10 Integration Techniques TMJC 2024 Page 1 of 24 H2 Mathematics (9758) Chapter 10 Integration Techniques Discussion Questions (Solutions) Level 1 Integration – Reverse of Differentiation 1 Find  d cosd   . Hence, find sin d   .  d cos cos sind        cos sin d cos cos d sin d cos sin d = cos d cos sin cos ', where ' C C C C C C                                         Integration of Standard Functions 2 Find the following integrals: (a)   5 32 e 3e dx x x (b) 1 2 21 d k xx       , 0k  (c)   2 5 2 dx x x x    Q2 Solutions (a) 5 3(2 e 3e ) dx x x 1 5 32 1 5 32 5 32 (2e 3e ) d 2e 3e 1 3 2 4e e x x x x x x x C C              Integration is reverse of differentiation

Chapter 10 Integration Techniques TMJC 2024 Page 2 of 24 (b)   2 1 1 2 1 2 4 41 d 1 d 44ln 4(1 4) 4ln 4 4ln 3 0 k k k x xx x x x x x k k k k k kk                                   (c) 2 3 1 1 2 2 2 (2 5)( 2) d 2 10 d 2 10 d x x x x x x x x x x x x            5 3 2 24 2 205 3x x x C    Need to expand first before integrating term by term.

Chapter 10 Integration Techniques TMJC 2024 Page 3 of 24 Integration involving the function and its derivative 3 Find the following integrals: (a) 23 7 dx x x (b) 2 d2 6 x xx  (c) 32 1e dxx x   Q3 Solution (a) 23 7 dx x x       1 2 2 3 2 2 3 2 2 1 14 3 7 d14 3 71 314 2 1 3 721 x x x x C x C              (b) 2 2 2 d2 6 1 4 d4 2 6 1 ln 2 64 x xx x xx x C         (c) 32 1e dxx x   3 3 2 1 1 1 3 e d3 1 e3 x x x x C       Formula to memorise (not in MF27) and apply   1 f ( ) f ( ) 1 f ( )(2) For , 1, f ( ) f ( ) d d ln f ( ) f ( ) (3) f ' ( ) e d e [f ( )](1) For , 1, f ' ( )[f ( )] d 1 x x n n xn n x x x x x C x x x C xn n x x x C n                     Note: 1. 22 6 x x  is a proper function 2.      f ' d ln f , f x x x C Cx     Remember to put modulus

Chapter 10 Integration Techniques TMJC 2024 Page 4 of 24 Integration of Rational Algebraic Functions (including MF27) 4 Find the following integrals: (a) 2 1 d3 4 tt (b)    1 d3 4 xx x  (c) 2 10 d 2 11 x x x  Q4 Solutions (a)     2 2 2 1 d3 4 1 d 3 2 1 1 3 2 ln22 3 3 2 3 3 2ln12 3 2 tt t t t C t t C t                      (b) 1 d( 3)( 4) xx x  1 1 d( 3) ( 4) ln 3 ln 4 3ln 4 xx x x x C x Cx              (c) 2 10 d 2 11 x x x    2 1 1 110 d 1 10 10 1tan 10 10 110 tan 10 x x x C x C                  

Chapter 10 Integration Techniques TMJC 2024 Page 5 of 24 Integration of Trigonometric Functions 5 (a) 3sin cos dx x x (b) 2sin dx x Q5 S