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 antideriv