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Integration - CHAPTER 10 Integration Integration can be...

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CHAPTER 10 Integration Integration can be thought of as the opposite of differentiation but is also a method for finding the area under a graph . It is an important mathematical technique, which will be familiar if you have done A-level maths. In this chapter we look at techniques for integrating standard functions, including integration by substitution and by parts and at economic applications including calculation of consumer surplus . 1. The Reverse of Differentiation If we have a function y ( x ), we know how to find its derivative dy dx by the process of differen- tiation . For example: y ( x ) = 3 x 2 + 4 x - 1 dy dx = 6 x + 4 Integration is the reverse process: When you know the derivative of a function, dy dx , the process of finding the original function, y , is called integration. For example: dy dx = 10 x - 3 y ( x ) = ? If you think about how differentiation works, you can probably see that the answer could be: y = 5 x 2 - 3 x However, there are many other possibilities; it could be y = 5 x 2 - 3 x +1, or y = 5 x 2 - 3 x - 20, or . . . in fact it could be any function of the form: y = 5 x 2 - 3 x + c where c is a constant. We say that (5 x 2 - 3 x + c ) is the integral of (10 x - 3) and write this as: The with integral respect of to x (10 x - 3) dx = 5 x 2 - 3 x + c 173
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174 10. INTEGRATION c is referred to as an “arbitrary constant” or a “constant of integration”. More generally: Integration is the reverse of differentiation. If f ( x ) is the derivative of a function f ( x ), then the integral of f ( x ) is f ( x ) (plus an arbitrary constant): f ( x ) dx = f ( x ) + c 1.1. Integrating Powers and Polynomials In the example above you can see that since differentiating powers of x involves reducing the power by 1, integrating powers of x must involve increasing the power by 1. The rule is: Integrating Powers of x : x n dx = 1 n + 1 x n +1 + c ( n = - 1) It is easy to check that this rule works by differentiating: d dx 1 n + 1 x n +1 + c = x n You can also see from this that the rule doesn’t work when n = - 1. But it works for other negative powers, for zero, and for non-integer powers – see the examples below. We can apply this rule to integrate polynomials. For example: (4 x 2 + 6 x - 3) dx = (4 x 2 + 6 x - 3 x 0 ) dx = 4 × 1 3 x 3 + 6 × 1 2 x 2 - 3 x 1 + c = 4 3 x 3 + 3 x 2 - 3 x + c It is easy to make mistakes when integrating. You should always check your answer by dif- ferentiating it to make sure that you obtain the original function. Examples 1.1 : Integrating Powers and Polynomials (i) What is the integral of x 4 - 2 x + 5? ( x 4 - 2 x + 5) dx = 1 5 x 5 - x 2 + 5 x + c (ii) Integrate 2 - t 5 5 . 2 - t 5 5 dt = 2 t - t 6 30 + c
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10. INTEGRATION 175 (iii) If dy dx = (2 - x )(4 - 3 x ), what is y ? y = (2 - x )(4 - 3 x ) dx = (8 - 10 x + 3 x 2 ) dx = 8 x - 5 x 2 + x 3 + c (iv) Integrate 1 + 10 z 3 . 1 + 10 z 3 dz = ( 1 + 10 z - 3 ) dz = z + 10 × 1 - 2 z - 2 + c = z - 5 z 2 + c (v) If f ( x ) = 3 x what is f ( x )? f ( x ) = 3 xdx = 3 x 1 2 dx = 3 × 2 3 x 3 2 + c = 2 x 3 2 + c (vi) Integrate the function 3 ax 2 + 2 tx with respect to x .
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