RI 2022 C8A Integration Techniques

RAFFLES INSTITUTION H2 Mathematics (9758) 2022 Year 5 ______________________________ Chapter 8A: Integration Techniques Page 1 of 23 Chapter 8A: Integration Techniques SYLLABUS INCLUDES  Integration of , , , , , , and , , , and  Integration by a given substitution  Integration by parts PRE-REQUISITES  Basic Trigonometry  Partial Fractions  Differentiation Techniques CONTENT 1. Introduction 1.1 Integration as the Reve rse Process of Differentiation 1.2 Indefinite Integral 1.3 Definite Integral 2. Integration of Standard Functions 2.1 Integration of the form nx and Variations, where 1n 2.2 Integration of the form 1x and Variations 2.3 Integration of Exponentia l Function and Variations 2.4 Integration of Standard Trig onometric Functions and Variations 2.4.1 Integration using Trigonometric Identities 3. Other Integration Techniques 3.1 Integration of 22 1 ax , 22 1 ax 3.2 Integration Involving Partial Fractions 3.2.1 Integration of and 3.3 Integration by Substitution or Change of Variable 3.4 Integration by Parts f' ( ) f( ) n x x f( )f' ( ) e xx 2sin x 2cos x 2tan x sin cosmx nx cos cosmx nx sin sinmx nx 22 1 ax 22 1 ax 22 1 ax 22 1 x a 22 1 ax 22 1 x a You do not need to print this set of lecture notes as it will be included (with solutions) in the next mathematics package (Book 5) issued in Year 6.

Raffles Institution H2 Mathematics 2022 Year 5 _______________________________________________________________________________________________________ _____________________________ Chapter 8A: Integration Techniques Page 2 of 23 1 INTRODUCTION We have seen that differentiating a function with respect to its variable gives us its derivative, i.e. the function that tells us the instantaneous rate of change of the original function with respect to its variable. This raises the question of whether it is possible to recover the original function, given its derivative. Just as differentiation is a process that gives you the derivative of a function, integration as introduced here will be a process that seeks to give you the original function given it s derivative, albeit wi th some additional information required. Here, integration is viewed as anti-differentiation, and the functions obtained after integration are known as anti-derivatives. In the next chapter, we will then see how the processes of differentiation and integration are linked via the fundamental ideas of limits and use these principles to show how, just as differentiation may be used to find gradients of curves, integration may also be used to find areas and volumes. 1.1 Integration as the Revers e Process of Differentiation One basic way to perform integration is to modi fy the integrand (the expression that is to be integrated), such that it represents the derivative of a common function. The ability to integrate a function in such a manner depends primarily on recognizing it as the derivative of another function. Exercise : Given the following expression for f( )x , find an expression F( )x such that d F( ) f ( )d x xx  . f( )x F( )x f( )x F( )x 1 x e x e x 2x 2x cos x sinx 23x 3x 2sec x tanx 3x 4 4 x 1 x lnx

Raffles Institution H2 Mathematics 2022 Year 5 _______________________________________________________________________________________________________ _____________________________ Chapter 8A: Integration Techniques Page 3 of 23 1.2 Indefinite Integral 1.2.1 Notation Suppose f and F are two functions related as follows: d d ( F () ) f () x x x . Then, for any constant c, d d F( ) f ( ) x x cx . Hence, we have F( ) f ( )dx cx x  . We call f( ) dx x the indefinite integral of f with respect to x , c the arbitrary constant of integration, f( )x the integrand,  the integral sign. For example, we have 2d d 1 (3 )3 32x x x   and 2d d 1 (3 )3 0 0 32x x x    , therefore 21(3 ) d (3 ) 2x xx c   . 1.2.2 Basic Rules of Integration (i)  f () g ()d f () d g () dx xx x x x x   (ii) f( ) d f( ) dkx x k x x , where k is a constant, 0k (iii) d f () d f ()d x xx cx  (iv)   d f( ) d f( )d x xxx  Note :  f () g ()d f () d g () dx xx x x x x   f( ) df( ) dg( ) g( ) d x xx xx x x   

Raffles Institution H2 Mathematics 2022 Year 5 _______________________________________________________________________________________________________ _____________________________ Chapter 8A: Integration Techniques Page 4 of 23 1.3 Definite Integral If f is a function defined and continuous on an interval I = ,uv and f( ) d F ( )x xx c , then for a, b  I such that uabv , and the definite integral of f( )x from a to b w.r.t. x is denoted by  f( ) d F ( ) F ( ) F ( ) b b aa x xx ba  where a and b are called the lower and upper limits of the integral respectively. 1.3.1 Basic Properties of Definite Integrals (i) f( ) d 0 a a xx  (ii) f( ) d f( ) d ba ab x xx x (iii) f () d f () d f () d bc b aa c x xx xx x , where c is such that acb (iv)  f () g ()d f () d g () d bb b aa a x xx x x x x   (v) f( ) d f( ) d bb aa kx x k x x , where k is any constant, 0k 2 Integration of Standard Functions 2.1 Integration of the form nx and Variations, where 1n Recall that 1d (1 )d nnx nxx   and    1d f( 1 ) f f 'd nn x nx xx     . For example,  5d d xx =  4 5 x  5d 25d xx  =    4 52 5 2x  f2 5 , f ' 2 , 4xx x n   1 d, 1 1 n n xxx c n n        1 ff ' f d , 1 1 n n xxxx c n n      Note: In particular, when f x ax b ,  \0,ab are constants, then f ' x a and   11() ()dd , 1 1( 1 ) nn nn ax b ax baa x b x c a x b x d nna n     

Raffles Institution H2 Mathematics 2022 Year 5 _______________________________________________________________________________________________________ _____________________________ Chapter 8A: Integration Techniques Page 5 of 23 Example 1 Find (a) 3 1 dx xx     1 23 dx xx 3 22 3 22 xx c    3 2 2 12 23 x cx   (b)  2 54 dx x  21 55 4 d5 x x   3 541 53 x c  31 5415 x c   f5 4xx  f ' 5x  2n Check: Check: 3 2 23 d1 2 1 d23 x cxxx x      33d1 54 54d1 5 xc xx    (c)  3223 dx xx  321 42 3 d4 x xx    42231 44 x c    421 2316 x c   2f2 3 f( ) 4, 3 xx xx n    (d) 24 23 d(3 ) x xxx     42(2 3) 3 dx xx x    32 1 33 c xx      2f3 x xx f ' 2 3x x 4n Check: Check:  422 3d1 23 ( 23 )d1 6 xc x xx     3 242 d1 2 3 d( 3 ) 33 xcx xxxx     2.2 Integration of the form 1x and Variations  ln , 0ln ln , 0, xxx xx    d1ln , 0d dln d1 1d ln , 0.d xxxxxx xxxx x        When the domain is not specified, we write xx d 1  = ln x + c .

Raffles Institution H2 Mathematics 2022 Year 5 _______________________________________________________________________________________________________ _____________________________ Chapter 8A: Integration Techniques Page 6 of 23 In general, we have    f ' dl n ff x x xcx  . Example 2 Find (a) 3 dxx 13 d xx  3ln x c (b) 1 d15 xx 15 d515 xx   1 ln 1 55 x c   f1 5x x f ' 5x  (c)  2 21 d 22 1 x x xx    2 14 2 d2 22 1 x x xx   21 ln 2 2 12 x xc     2f2 2 1x xx  f ' 4 2x