TMJC H2 Chapter 10 Integration Techniques Learning Package 2024

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Chapter 10 Integration Techniques TMJC 2024 Page 1 of 29 H2 Mathematics (9758) Chapter 10 Integration Techniques Core Concept Notes Success Criteria: Surface Learning Deep Learning Transfer Learning  Integrate derivatives to obtain the anti-derivatives / integrals of a function (i.e. indefinite integration is the reverse process of differentiation)  Evaluate definite integrals using graphing calculator  Integrate the following standard functions: ➢ Constant, a ➢ nx ,  \1n− ➢ 1x− ➢ ex ➢ xa  Recognise and integrate integrands of the form  f ( ) f ( ) n xx , where n  Recognise and integrate integrands of the form f ( )f '( )e xx  Integrate basic trigonometric functions  Evaluate definite integrals using anti-derivatives: f ( ) d F( ) F( ) b a x x b a=− , where d F( ) f ( )d xxx =  Integrate powers of basic trigonometric functions  Use MF27 integral formulas to integrate functions of the following standard forms: (a) 22 1 ax+ (b) 22 1 ax− (c) ( ) 22 1 ,ax xa −  (d) ( ) 22 1 ,xa xa −   Recognise and integrate integrands of the form f ( ) dg( ) x xx or f ( ) d g( ) x x x  Use a given substitution to simplify and integrate an expression  Use integration by parts to integrate an expression

Chapter 10 Integration Techniques TMJC 2024 Page 2 of 29 §1 Indefinite Integrals If two functions F(x) and f(x) are related as follows: ( )d F( ) f ( )d xxx = , then f(x) is the derivative of F(x) and F(x) is called an anti-derivative or integral of f(x). Illustration: Since 2d ( 1) 2d xxx += , 2 1x + is an anti-derivative of 2x. Since 2d ( 7) 2d xxx −= , 2 7x − is an anti-derivative of 2x. Since 2d ( ) 2d xxx = , 2x is also an anti-derivative of 2x. Observe that   ( )d d dF( ) F( ) f ( ) 0 f ( )d d d x C x C x xx x x + = + = + = where C is any arbitrary real constant. Recall that the process of finding f ( ) dxx for a given function f is called integration. We know that f ( ) dxx is actually equivalent to F(x) + C, the collection of all anti-derivatives of f(x), and we write: Note: Integration is the “reverse” of differentiation. i.e.: f ( )d F( )x x x C=+ because ( )d F( ) f ( )d x C xx += Example: (1) cos d sinx x x C=+ because ( )d sin cosd x C xx += (2) ( ) ( ) 9 10 15 d 5 10x x x C− = − + because ( ) ( ) ( ) 10 9 9d 1 10 5 5 5d 10 10 x C x xx  − + = − = − (3) 51e dx x+  = 511 e5 x C+ + because 5 1 5 1 5 1d 1 5 e e ed