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Unformatted text preview: Basic Mathematics Indefinite Integration R Horan & M Lavelle The aim of this package is to provide a short self assessment programme for students who want to be able to calculate basic indefinite integrals. Copyright c 2004 [email protected] , [email protected] Last Revision Date: June 7, 2004 Version 1.0 Table of Contents 1. AntiDerivatives 2. Indefinite Integral Notation 3. Fixing Integration Constants 4. Final Quiz Solutions to Exercises Solutions to Quizzes The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials. Section 1: AntiDerivatives 3 1. AntiDerivatives If f = dF dx , we call F the antiderivative (or indefinite integral ) of f . Example 1 If f ( x ) = x , we can find its antiderivative by realising that for F ( x ) = 1 2 x 2 dF dx = d dx ( 1 2 x 2 ) = 1 2 × 2 x = x = f ( x ) Thus F ( x ) = 1 2 x 2 is an antiderivative of f ( x ) = x . However, if C is a constant: d dx ( 1 2 x 2 + C ) = 1 2 × 2 x = x since the derivative of a constant is zero. The general antiderivative of x is thus 1 2 x 2 + C where C can be any constant. Note that you should always check an antiderivative F by differenti ating it and seeing that you recover f . Section 1: AntiDerivatives 4 Quiz Using d ( x n ) dx = nx n 1 , select an antiderivative of x 6 (a) 6 x 5 (b) 1 5 x 5 (c) 1 7 x 7 (d) 1 6 x 7 In general the antiderivative or integral of x n is: If f ( x ) = x n , then F ( x ) = 1 n + 1 x n +1 for n 6 = 1 N.B. this rule does not apply to 1 /x = x 1 . Since the derivative of ln( x ) is 1 /x , the antiderivative of 1 /x is ln( x ) – see later. Also note that since 1 = x , the rule says that the antiderivative of 1 is x . This is correct since the derivative of x is 1. Section 1: AntiDerivatives 5 We will now introduce two important properties of integrals , which follow from the corresponding rules for derivatives. If a is any constant and F ( x ) is the antiderivative of f ( x ), then d dx ( aF ( x )) = a d dx F ( x ) = af ( x ) . Thus aF ( x ) is the antiderivative of af ( x ) Quiz Use this property to select the general antiderivative of 3 x 1 2 from the choices below. (a) 2 x 3 2 + C (b) 3 2 x 1 2 + C (c) 9 2 x 3 2 + C (d) 6 √ x + C Section 1: AntiDerivatives 6 If dF dx = f ( x ) and dG dx = g ( x ) , from the sum rule of differentiation d dx ( F + G ) = d dx F + d dx G = f ( x ) + g ( x ) . (See the package on the product and quotient rules. ) This leads to the sum rule for integration : If F ( x ) is the antiderivative of f ( x ) and G ( x ) is the antiderivative of g ( x ), then F ( x ) + G ( x ) is the antiderivative of f ( x ) + g ( x )....
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 Spring '11
 YucelTANDOGDU
 Calculus, Derivative, Fundamental Theorem Of Calculus, dx, antiderivatives

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