ACJC 2026 Integration Techniques Lecture Notes

10 INTEGRATION TECHNIQUES SYLLABUS ▪ Integration of o ( )f ( ) f ( ) n xx (including 1n=− ) o f ( )f ( )e xx o 2 2 2sin , cos , tanx x x o 2 2 2 2 2 2 22 1 1 1 1, , , a x a x x a ax+ − − − ▪ Integration by a given substitution ▪ Integration by parts

ACJC 2025/26 H2 Mathematics (9758) 2 CONTENTS 1 Integration as a Reverse Operation to Differentiation ...................... 3 2 Integration of Standard Functions .................................................... 3 2.1 Integration of ( )f ( ) f ( ) n xx ..................................................... 5 2.2 Integration of Exponential Functions ...................................... 7 2.3 Integration of Trigonometric Functions .................................. 8 2.3.1 Integration of Standard Trigonometric Functions ......... 8 2.3.2 Integration using Trigonometric Identities .................. 10 2.4 Integration of () () Px Qx or () () Px Qx ........................................... 13 2.4.1 Integration of 22 1 ax+ and 22 1 , xa ax  − ............ 13 2.4.2 Integration of 22 1 xa− and 22 1 ax− ............................ 15 2.4.3 Integration of 2 , 0px q aax bx c + ++ ............................... 16 2.4.4 Integration of 2 , 0, 0px q pa ax bx c +  ++ ................. 17 3 Integration by Substitution ............................................................ 20 4 Integration by Parts ........................................................................ 22 Annex A: Summary of Formulae ......................................................... 25 Annex B: Practice Questions on Integration Techniques ...................... 27

10 Integration Techniques 3 LECTURE 1 Lesson Outline • Integration as a Reverse Operation to Differentiation • Integration of Standard Functions - ( )f ( ) f ( ) n xx - Exponential Functions - Trigonometric Functions 1 INTEGRATION AS A REVERSE OPERATION TO DIFFERENTIATION Integration reverses the process of differentiation, i.e., for any constant C, ( )d f ( ) f ( ) f ( ) d f ( )d x x x x x Cx =  = + . For example, ( ) sin sin sin sind e cos e cos e d ed x x x x x x x Cx =  = +  . Notes • ( )d f ( )d xx means ‘differentiating the function f ( )x with respect to (w.r.t.) x’. • f ( ) d xx means ‘integrating the function f ( )x w.r.t o x’. • Integration is distributive with respect to addition, subtra ction and scalar multiplication, i.e., ( ).f ( ) .g( ) d f ( )d g( )da x b x x a x x b x x =    . 2 INTEGRATION OF STANDARD FUNCTIONS Recap integration of nx and ( ) , 0 n ax b a+ from ‘O’ level: 1 d , 1 1 n n xx x C n n + = + −+ 1 1d d lnx x x x C x − = = + 1()( ) d , 1 ( 1) n n ax bax b x C n an +++ = + − + 1 1( ) d d ln ax b x x ax b ax b Ca −+= + +=+ 

ACJC 2025/26 H2 Mathematics (9758) 4 Example 1 Integrate the following with respect to x. (a) ( ) 4 3 475 21 x x x −+ + (b) (1 )(2 1) 2 xx x −− (c) 1 2xx+− Solution (a) ( ) ( ) ( )( ) 25 4 3 214 7 55 d 4ln 7 5 2 221 xxx x x Cx x − +− + = − + + −+   ( ) 5 2 74ln 4 2 1 x x C x = − − + + (b) 2(1 )(2 1) 1 1 3 2d d 22 x x x x xx xx − − − + − =    3 2 35 22 35 22 11 3 2 d2 1 3 2 1 3 52 2 2 2 2 5 x x x x x x x C x x x C = − + −  = − + − +   =− + − +   (c) 1 1 2 dd 2 2 2 xxxx x x x x x x ++= + − + − + +    ( ) ( ) 33 3322 22 21 d 2 d( 2) ( ) 2 21 1 1 2332 3 3 22 xx x x x xxx x x C x x C ++= = + ++−   += + + = + + +      ■

10 Integration Techniques 5 2.1 Integration of ( )f ( ) f( ) n xx From ( ) ( ) 1d f ( ) ( 1) f ( ) f ( ), 1d nn x n x x nx + = + − and ( )d f ( )ln f ( )d f ( ) xxxx = , integrate both sides with respect to x to get For 1n− ( ) ( ) 1 f ( ) f ( ) f ( ) d 1 n n xx x x C n +  =+ + For 1n=− ( ) 1 f ( ) f ( ) f ( ) d d ln f ( ) f ( ) xx x x x x C x −  = = + Examples of ( ) ( ) 1 f( )f ( ) f( ) d 1 n n xx x x C n +  =+ + where − 1n (a)    24 23 f ( ) f ( ) ( 1)(2 ) ( 1) d 4 nx x xx x x C  ++ = +   (b)    22 23 f ( ) f ( ) ( 1)(2 )( 1) d 2 nx x xx x x C − −  ++ = + −    (c)   3 2 f ( ) f ( ) (sin ) cos sin d 3nx x xx x x C  =+  (d)   3 22 f ( ) f ( ) (tan )sec tan d 3nx x xx x x C  =+  Examples of ( ) 1 f ( )f ( ) f( ) d d ln f( ) f( ) xx x x x x C x −  = = + (a) ( ) 2 2 2 d ln 11 x x x Cx = + ++  (b) 2 32 32 32 d ln 11 xx x x x Cxx + = + + +++  (c) coscot d d ln sinsin xx x x x C x= = + (d) sintan d d cos 1ln cos ln cos ln sec xx x x x x C C x xC −=− =− + = + =+  = = 22 22 sin (sin ) tan (tan ) xx xx

ACJC 2025/26 H2 Mathematics (9758) 6 Example 2 Find (a) 2 3 7 d 1 x x x +   (b) 1 dln xxx  (c) 1 d1 4e x x−−  Solution (a) 2 3 7 d 1 x x x +   ( ) 1 23 27 1 dx x x − =+ ( ) ( ) 1 23 2 f ( ) f ( ) 7 1 d nx x x x x −  =+     ( ) 1 3 2 317 14 1133 2 x C x C + =  + = + + (b) 1 dln xxx  11 dln xxx=   f ( ) f ( ) d ln ln ln x x x x C x  = = +      (c) 1 d1 4e x x−−  ( ) 1 d e e 4xx x−= −   f ( ) f ( ) d ln e 4e4 x x x x xC  = = − +−      ■ Use   f ( ) df( ) x xx =+ln f( )xC Use   f ( ) df( ) x xx =+ln f( )xC Use ( )f ( ) f( ) d n x x x ( ) + = + 1 f( ) 1 n x n

10 Integration Techniques 7 2.2 Integration of Exponential Functions Based on the differentiation of ex and ,0xaa  , we have the following results: 1e d eax b ax bxC a ++ =+ and 1d , 0,1ln xxa x a C a a= +  . Differentiate f ( )e x and f ( )xa respectively , we have the following results: e d exx xC=+ f ( ) f ( )f ( )e d exxx x C =+ 1d ln xxa x a C a=+ f ( ) f ( ) 1f ( ) d ln  =+ xxx a x a C a Examples of  =+ f( ) f( )f ( )e d exxx x C (a)  f ( ) 22 f ( ) 2 e d e x xx x xC     =+ (b)  f ( ) 22 f ( ) 2 e d e x xx x x x C      =+ (c)  f ( ) sin sin f ( ) cos e d e x xx x x x C      =+ (d) f ( ) 2 tan tan f ( ) sec e d e x xx x x x C      =+  Example 3 Integrate the following with respect to x. (a) 534 x− (b) sin(sin cos )e xxx x x+ (c) tane (1 sin )(1 sin ) x xx+−

ACJC 2025/26 H2 Mathematics (9758) 8 Solution (a) 534 dx x−  ( ) 53 f ( ) 1 3 4 d3 x x x−  =−−  ( ) ( ) 53 53 11 43 ln 4 1 43ln 4 x x C C − − =+ −  =− + (b) f ( ) sin f ( ) (sin cos ) e d x xx x x x x x  +=    sinexx C+ (c) tane d(1 sin )(1 sin ) x xxx+−   tan tan 22 ee d d1 sin cos xx xxxx== −  f ( ) 2 tan f ( ) tan sec e d e x x x x xx C  = =+     2.3 Integration of Trigonometric Functions 2.3.1 Integration of Standard Trigonometric Functions Recap integration of trigonometric functions from O-level: cos d sinx x x C=+ ( ) ( )1cos d sin , 0ax b x ax b C aa+ = + +  sin d cosx x x C=− + ( ) ( )1sin d cos , 0ax b x ax b C aa+ =− + +  2sec d tanx x x C=+ ( ) ( ) 2 1sec d tan , 0ax b x ax b C aa+ = + +  Note that the angle x is measured in radians. Also recap differentiation of trigonometric functions from Chapter 5: 2cosec d cot=− + x x x C ( ) ( ) 2 1cosec d cot+ =− + + ax b x ax b C a sec tan d sec =+ x x x x C ( ) ( ) ( )1sec tan d sec+ + = + + ax b ax b x ax b C a cosec cot d cosec =− + x x x x C ( ) ( ) ( )1cosec cot d cosec+ + =− + + ax b ax b x ax b C a Use  f( )f ( ) d xx a x =+ f( )1 ln x a aC Use  f( )f ( )e dxxx =+ f( )e x C Use  f( )f ( )e dxxx =+ f( )e x C ■

10 Integration Techniques 9 Note Other integration formulae of trigonometric functions which can be found in MF 27, are given as follows without the modulus, you should add in the modulus sign where it is necessary: ( ) ( )1 2tan d ln sec π= +  x x x C x ( ) ( )cot d ln sin 0 π= +   x x x C x (