TMJC Term 4 Revision Chapter 10 Techniques of Integration Solutions

Term 4 Revision Topical Quick Check: Integration Techniques TMJC 2024 Page 1 of 6 JC1 H2 Mathematics (9758) Term 4 Revision Topical Quick Check Chapter 10 Integration Techniques 1 HCI Promo 9758/2022/Q8 (a) Find ( ) 13 tan 3 dt t t −  . [4] (b) Using the substitution 2 1ux=+ , show that ( ) 17 32 3 0 1dx x x + can be expressed as 41 331 d2 b a u u u− , where a and b are constants to be determined. Hence find the exact value of ( ) 17 32 3 0 1dx x x + . [5]

Term 4 Revision Topical Quick Check: Integration Techniques TMJC 2024 Page 2 of 6 1 HCI Promo 9758/2022/Q8 (a) Find ( ) 13 tan 3 dt t t −  . [4] 1 (a) ( ) 13 tan 3 dt t t −  ( ) ( ) 22 1 2 3 3 3tan 3 d22 13 tt tt t −=− + ( ) 22 1 2 39 tan 3 d22 19 tt tt t −=− + ( ) ( ) 2 1 2 11tan 3 1 d22 19 t tt t −= − − + ( ) ( ) ( ) 2 113 1 1tan 3 tan 32 2 2 3 t t t t C−−= − + + ( ) ( ) 2 1 13 1 1tan 3 tan 32 2 6t t t t C−−= − + + (b) Using the substitution 2 1ux=+ , show that ( ) 17 32 3 0 1dx x x + can be expressed as 41 331 d2 b a u u u− , where a and b are constants to be determined. Hence find the exact value of ( ) 17 32 3 0 1dx x x + . [5] (b) Let 2 1ux=+ , then d 2d u xx = . When 0, 1xu== . When 7, 8xu== . ( ) 17 32 3 0 1dx x x + ( ) ( ) 17 22 3 0 1 1 2 d2 x x x x=+  ( )( ) 18 3 1 1 1d2 u u u=−  418 33 1 1 d2 u u u=−  (Shown) 874 33 1 1 3 3 2 7 4uu=−   ( ) ( ) 743 1 1 1 1222 7 4 7 4 = − − + 1209 56=

Term 4 Revision Topical Quick Check: Integration Techniques TMJC 2024 Page 3 of 6 2 EJC Promo 9758/2022/Q6 (a) Find 231e d .xxx +  [1] (b) Find ( ) 2sin 5 d .xx [3] (c) Find 2 .41 d47 x xxx −+  [5]

Term 4 Revision Topical Quick Check: Integration Techniques TMJC 2024 Page 4 of 6 2 EJC Promo 9758/2022/Q6 (a) Find 231e d .xxx +  [1] 2 (a) ( ) 22 2 3 1 3 1 31 1e d 6 e d 6 1 e6 xx x x x x x c ++ + = =+  (b) Find ( ) 2sin 5 d .xx [3] (b) ( ) 2 1 cos10sin 5 d d 2 1 1 sin10 2 2 10 11 sin102 20 xx x x xxC x x C −= = − +  = − +  (c) Find 2 .41 d47 x xxx −+  [5] (c) ( ) ( ) ( ) ( ) ( ) 22 2 2 2 2 21 21 2 118482 4 1 4 1 1 8 4 1 1 8 4 1 2 4 1 1 1 1ln 4 4 1782 2 1 4 1 1 1 1 2 1ln 4 4 17 tan8 2 2 dd4 7 4 7 dd4 n 44 1 1 2 1l 4 4 17 tan8 16 4 7 4 7 d xx xx x xx xx x xx x C xx x C xxxx xxxx x − − −+ − + − + − − + − + −+ −+  −− +  +   = =+ =+ =  + +  − = − + +        

Term 4 Revision Topical Quick Check: Integration Techniques TMJC 2024 Page 5 of 6 3 MI PU2 P1 Promo 9758/2022/Q4 (i) Find cos 2 sin dx x x . [3] (ii) Find 1sin 2 2 e d 14 x x x − −   . [2] (iii) Find 2 5 d6 13 xxx++  . [3] 3(i) ( ) ( ) 2 2 3 cos 2 sin d 2cos 1 sin d 2sin cos sin d 2 cos cos3 x x x x x x x x x x x x C =− =− =− + +   Alternative Method ( )1cos 2 sin d sin 3 sin d2 1 cos3 cos23 11cos3 cos62 x x x x x x x xc x x c =− −= + + =− + +  3(ii) ( ) 11 1 sin 2 sin 2 22 sin 2 1 1 2 e d e d 214 12 1 e2 xx x xx x x c −− − = − − =+    3(iii) ( ) ( ) 22 2 2 2 1 55 dd6 13 3 3 13 5 d 32 53tan22 xxxx x x x x c− =++ + − + = ++ +=+      

Term 4 Revision Topical Quick Check: Integration Techniques TMJC 2024 Page 6 of 6 Answer Key No. Year JC Answers 1 2022 HCI (a) ( ) ( ) 2 1 13 1 1tan 3 tan 32 2 6t t t t C−− − + + (b) 1209 56 2 2022 EJC (a) 2 131 e6 x c+ + (b) 11 sin102 20x x c−+ (c) ( ) 211 1 2 1ln 4 tan8 1 4 7 641 x cx x − −+ +  − + 3 2022 MI (i) 32 cos cos3 x x C− + + 11or cos3 cos62 x x c − + + (ii) 1sin 21 e2 x c − + (iii) 153tan22 x c− + +