MTH
mth.122.handout.09

# 7 if the limit exists we say its is convergent if the

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7 If the limit exists we say its is convergent , if the limit does not exists we say it is divergent . 4 I’m using integration by parts and this will be reviewed in class. 5 This limit will be reviewed in class. 6 Finite number 7 Finite number 2

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3. If f has a discontinuity at c , where a < c < b , and both Z c a f ( x ) d x and Z b c f ( x ) d x are convergent, then we define Z b a f ( x ) d x = Z c a f ( x ) d x + Z b c f ( x ) d x 2.1 Example 1. Evaluate. Z 5 2 1 x - 2 d x Work: The integral is improper because the integrand has a vertical asymptote at x = 2. Z 5 2 1 x - 2 d x = lim t 2 + Z 5 t 1 x - 2 d x = lim t 2 + 2 x - 2 5 t = lim t 2 + 2 3 - t - 2 = 2 3 3 Comparison Theorems Comparison Theorem: Suppose f and g are continuous functions with f ( x ) g ( x ) 0 for x a . 1. If Z a f ( x ) d x is convergent, then Z a g ( x ) d x is convergent. 2. If Z a g ( x ) d x is divergent, then Z a f ( x ) d x is divergent. It is useful to note that we often use the following integral for comparison purposes: Z 1 1 x p d x. Here, it can be shown that this integral is convergent if p > 1 and divergent if p 1. 3
3.1 Example 1. Is the integral Z 1 1 + e - x x d x convergent or divergent?

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• Spring '10
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