02Indefinite Integration as Antidifferentiation2

02Indefinite Integration as Antidifferentiation2 - ( 29...

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I NDEFINITE I NTEGRATION AS A NTI -D IFFERENTIATION A function F is called an antiderivative of a function f on a given interval I if ) ( ) ( x f x F = for all x on the interval. Ex . 1) 3 3 1 ) ( x x F = is an antiderivative of 2 x ) ( ) ( ' 2 x f x x F = = However, this is not the only antiderivative of ) ( x f - eg. 5 3 1 , 2 3 1 , 3 1 3 3 3 - + x x x are all antiderivatives of 2 ) ( x x f = . The process of finding an antiderivative is called integration . C is called the constant of integration. Notation : + = C x F dx x f ) ( ) ( + = C x dx x 3 2 3 1 This is called the indefinite integral. Power Rule : (add one to the exponent and divide by new exponent) For 1 n : C n x x n n + + = + 1 1 Ex . 2) Find a) 3 x dx = b) 5 1 dx x = c) xdx = But: 1 - = n does not fit the pattern: + = C x dx x ln 1 C x dx + = 1
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Properties of the Indefinite Integral a) A constant factor can move to the outside of the integrand sign. + = = C x cF dx x f c dx x cf ) ( ) ( ) ( b) An antiderivative of a sum is the sum of the antiderivatives. ( 29 + + = + = + C x G x F dx x g dx x f dx x g x f ) ( ) ( ) ( ) ( ) ( ) ( c) An antiderivative of a difference is the difference of the antiderivatives.
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Unformatted text preview: ( 29 +-=-=-C x G x F dx x g dx x f dx x g x f ) ( ) ( ) ( ) ( ) ( ) ( Trigonometric Integration + = C x xdx sin cos +-= C x xdx cos sin Logarithmic Integration + = C e dx e x x Ex . 3) Find 2 3 2 x x dx +-= Ex . 4) Find 4cos xdx = Integration by Substitution For composite functions we can use substitution to make integration simpler. Ex . 5) Find -dx x ) 2 5 sin( Try the following examples: 4 3 t t e dt = dx x x 2 3 1 sin = 2 3 1 3 x dx x x + + = Note: In general: + = + = +-= C e k dx e C kx k dx kx C kx k dx kx kx kx 1 ) sin( 1 ) cos( ) cos( 1 ) sin( Assign: p. 363 # 9, 15, 17, 20, 21, 28; p.371 # 1, 4ad, 5ab, 6, 11, 12, 25, 27; Integration Worksheet...
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02Indefinite Integration as Antidifferentiation2 - ( 29...

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