E to write everything down with only one equals sign

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two, i.e. to write everything down with only one equals sign, so we introduce a new mathematical symbol for the “indefinite” integral and write: 2x dx = x + c 2 where the integration symbol is an elongated S standing for “summation” because, as mentioned in the Introduction above and as will be seen below, integration can be thought of as the limit of a summation process. ! Generalising this, we define the indefinite integral thus: f(x) dx = F(x) + c where F' (x) = f (x) Examples: Using our knowledge of differentiation, we find that: (i) x dx = (ii) x dx = 2 (iii) x dx = 3 (iv) x dx = 4
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90 (v) a dx = (vi) ( a + bx ) dx = (vii) ( 2 + 3 x + 4x + 5x - 6x - 7x ) dx = 2 3 4 5 (viii) dx = x dx = -2 (ix) dx = x dx = -3 (x) dx = x dx = -1 (xi) dx = ( x ) dx = -1 (xii) exp ( x ) dx = e dx = x (xii) exp ( 2x + 3) dx = e dx = 2 x + 3
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91 34. Rules of Integration From the above examples and our knowledge of differentiation, we find the following rules of integration: a x dx = n (a + b x ) dx = n e dx = log x + c exp ( ax +b) dx = e dx = e + c ax+b ax+b TUTORIAL EXERCISES 40. Find the following indefinite integrals: (i) ( 5 - 2 x + 3x - 2x + 4x - 8x ) dx 2 3 4 5 (ii) ( 2 + 3 x + 4x + 5x - 6x - 7x ) dx 2 3 4 5 (iii) ( 3x + 9x + 2x - 4 x - 6 ) dx 4 3 2 (iv) dx (v) dx (vi) e dx 4 x - 7 41. A competitive firm produces output q > 0 and has a marginal cost function given by MC(q) = q - 6 q + 9. Find its short-run total cost function. 2
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