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indef_integrals

# indef_integrals - Basic Mathematics Indefinite Integration...

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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. Anti-Derivatives 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: Anti-Derivatives 3 1. Anti-Derivatives If f = dF dx , we call F the anti-derivative (or indefinite integral ) of f . Example 1 If f ( x ) = x , we can find its anti-derivative 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 anti-derivative 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 anti-derivative of x is thus 1 2 x 2 + C where C can be any constant. Note that you should always check an anti-derivative F by differenti- ating it and seeing that you recover f . Section 1: Anti-Derivatives 4 Quiz Using d ( x n ) dx = nx n- 1 , select an anti-derivative 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 anti-derivative 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 anti-derivative of 1 /x is ln( x ) – see later. Also note that since 1 = x , the rule says that the anti-derivative of 1 is x . This is correct since the derivative of x is 1. Section 1: Anti-Derivatives 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 anti-derivative of f ( x ), then d dx ( aF ( x )) = a d dx F ( x ) = af ( x ) . Thus aF ( x ) is the anti-derivative of af ( x ) Quiz Use this property to select the general anti-derivative 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: Anti-Derivatives 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 anti-derivative of f ( x ) and G ( x ) is the anti-derivative of g ( x ), then F ( x ) + G ( x ) is the anti-derivative of f ( x ) + g ( x )....
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indef_integrals - Basic Mathematics Indefinite Integration...

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