2024 JPJC Chapter 8 Techniques of Integration Notes

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 8 Techniques of Integration (Students’ version) / Pg 1 Chapter 8: Techniques of Integration Content Outline  Integration of    f ' f n x x   (including 1n ),   f( )f ' e xx , 2sin x , 2cos x , 2tan x , sin cos ,cos cos ,sin sinmx nx mx nx mx nx , 2 2 1 a x , 2 2 1 a x , 2 2 1 a x and 2 2 1 x a  Integration by a given substitution  Integration by parts  Evaluation of definite integrals  Finding the approximate value of a definite integral using a graphic calculator References  Websites 1) http://integrals.wolfram.com/ [The Wolfram Integrator allows you to type in a function and obtain the indefinite integral almost immediately- good for checking answers in Tutorials.] 2) http://archives.math.utk.edu/visual. calculus/4/index.html [Contains numerous interactive tutorials on indefinite integrals, definite integrals and animations.]  Books 1) Catherine Berry, Val Hanrahan, Roger Porkess. “MEI Structured Mathematics Mathematics – Pure Mathematics 3”, 2nd ed, Hodder & Stoughton. Call Number: 510 BER. 2) Perkins & Perkins, “Advanced Mathematics – A Pure Course”, Collins. Call Number: PER 510.000. 3) Hugh Neill, Douglas Quadling, “OCR – Pure Math ematics 3”, Cambridge University Press. Call Number 510 NEI.

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 8 Techniques of Integration (Students’ version) / Pg 2 Relevant Formulas Found in MF26 Integrals (Arbitrary constants are omitted; a denotes a positive constant.) f ( )x f( ) dx x 2 2 1 x a 11 tan x a a       22 1 xa  1sin x a       ( x a ) 22 1 ax  1 ln2 x a a x a      ( )x a 22 1 xa  1 ln2 a x a a x      ( x a ) tan x ln(sec )x ( 1 2x  ) cot x ln(sin )x ( 0 x   ) cos ecx ln(cosec cot )x x  ( 0 x   ) sec x ln(sec tan )x x ( 1 2x  ) Trigonometry sin( ) sin cos cos sinA B A B A B   cos( ) cos cos sin sinA B A B A B   tan tantan( ) 1 tan tan A BA B A B    sin 2 2sin cosA A A 2 2 2 2cos 2 cos sin 2cos 1 1 2sinA A A A A      2 2 tantan 2 1 tan AA A  1 1 2 2sin sin 2sin ( )cos ( )P Q P Q P Q    1 1 2 2sin sin 2cos ( )sin ( )P Q P Q P Q    1 1 2 2cos cos 2 cos ( ) cos ( )P Q P Q P Q    1 1 2 2cos cos 2sin ( )sin ( )P Q P Q P Q   

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 8 Techniques of Integration (Students’ version) / Pg 3 1. Antiderivatives and Indefinite Integrals Definition If the function f ( )x is the derivative of a function F(x) with respect to x on an interval I, i.e. d [F( )] f( )d x xx  , for every value of x in I, then the function F(x) is called an antiderivative of the function f(x). Note: Antiderivatives are NOT unique. For example, if  f 2x x , then 2 2 2 3, , x x x   are all antiderivatives of f . 1.1 The Indefinite Integral If F is an antiderivative of f, we write (in integral notation): f( ) d F( )x x x c  , where c is an arbitrary constant. We call f ( ) dx x the indefinite integral of f.  - the integral sign dx - (integrate) with respect to x f ( )x - the integrand The process of fi nding antiderivatives is called integration and is the reverse process of differentiation, that is: 1.1.1 Properties of the Indefinite Integral If the functions f and g are integrable on interval I , and k is a constant, (i) f ( ) d f ( ) dk x x k x x  (ii)  f ( ) g( ) d f ( ) d g( ) dx x x x x x x     integrating differentiating

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 8 Techniques of Integration (Students’ version) / Pg 4 1.2 The Definite Integral f ( ) d b a x x is referred to as the definite integral of f(x) from x a to x b , where a is the lower limit and b is the upper limit. If f is continuous on [a, b] and F is any antiderivative of f on [a, b], then      f d F( ) =F( ) F( ), where F( ) f d . b b aa x x x b a x x x    . e.g.   53 3 3 5 2 1 1 5 1 148 2 d 2 2(5) 2(1) 3 3 3 3 xx x x                        We can use GC to evaluate 5 2 1 (2 ) dx x as shown below: Using the Home Screen: 1. Press alpha then window. 2. Press 4 to select “4: fnInt(”. 3. Complete the expression and then press enter.

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 8 Techniques of Integration (Students’ version) / Pg 5 1.2.1 Properties of the Definite Integral If the functions f and g are integrable on [a, b], and k is a constant, (i) f ( ) d 0 a a x x  (ii) f ( ) d f ( ) d b a a b x x x x  Proof:        f d F( ) =F( ) F( ) F( ) F( ) f d b a b aa b x x x b a a b x x      (iii) f ( ) d f ( ) d b b a a k x x k x x  (iv)   f ( ) g( ) d f ( ) d g( ) d b b b a a a x x x x x x x     (v) f ( ) d f ( ) d f ( ) d b c b a a c x x x x x x    , where a c b  2. Standard Integrals 2.1 Integrals Involving Algebraic Functions ( ) , nax b n   Recall: 1d ( ) ( 1)d n nx n xx    , where 1n  1d (1 1 )d n nx xxn   11 1 d ( )d n nx xx n    Hence, 11 1d nnx x x c n    And similarly, 1lnd ( )dx x x 1 d ln x x cx  

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 8 Techniques of Integration (Students’ version) / Pg 6 In general, 1d [( ) ] ( 1) ( )d n nax b n a ax bx     , where 1n   1 ( ) ( ) d ( 1) n n ax bax b x c a n      and d [ln( )]d aax bx ax b     1 1 d lnx ax b cax b a    Summarizing the above results, we have In general 1. 1 d 1 n n xx x c n    , where 1n  1 ( ) ( ) d ( 1) n n ax bax b x c a n      , where 1n  2. 1 d lnx x cx   1 1 d lnx ax b cax b a    Example 1 Find the following integrals, without the use of a graphing calculator. (a) 3 (5 +2) dx x  (b) 1 2 0 (4 5 ) dx x  (c) 2 3 3 2 2 2 2 2 3 2 4 6 (2 ) d 2 3(2 ) 3(2)( ) ( ) d (8 12 6 ) d x x x x x x x x x x                 (d) 22 d x xx   (e) 2 d2 3 xx 

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 8 Techniques of Integration (Students’ version) / Pg 7 2.2 '' f ( )Integrals of the Form f ( )[f ( )] d and df( ) n xx x x x x  Recall:   1 1 d [f ( )] ( 1)[f ( )] f ( )d d [f ( )] [f ( )] f ( )d 1 n n n n x n x ' xx x x ' xx n           Hence, 1[f ( )] f ( )[f ( )] d 1 n n x' x x x c n     And recall that, d f ( )[ln f ( )]d f ( ) ' xxx x  Hence, f ( ) d ln f ( )f ( ) ' x x x cx   Summarizing the above results, 1. 1[f ( )] f ( )[f ( )] d 1 n n x' x x x c n     , where 1n  2. f ( ) d ln f ( )f ( ) ' x x x cx   Example 2 Find the following integrals. (a) 2 10 4 (2 5) dx x x   (b) 2 8 4 d 1 x x x x     (c) 2 3 3 d7 2 x xx  (d) 2 3 1 d 3 1 x xx x   

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 8 Techniques of Integration (Students’ version) / Pg 8 2.3 Integrals Involving Exponential Functions Recall: d (e ) ed x x x  Hence, e d ex x x c  In general, f ( ) f ( )d [e ] f ( )ed x x ' xx  Hence, f ( ) f ( ) f ( )e d ex x' x x c   In general 1. e d ex x x c  f ( ) f ( ) f ( )e d ex x' x x c   Example 3 Find the following integrals. (a) 1 2 e d x x (b) 24 1 2 e dxx x   (c) 10 deax x , where a is a constant. =

Jurong Pioneer Junior College H2 Mathematics (9758) JC1-2023 Chapter 8 Techniques of Integration (Students’ version) / Pg