sec7_8

sec7_8 - ln x dx Section 7.8 Improper Integrals 3 4....

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7.8 Improper Integrals 1. Infinite Limits of Integration One or both limits of integration are inFnite Defnition 1.1. We defne an improper integral using limits and say an improper integral is convergent iF the limit is a real number. Otherwise, it is divergent . ± 1 f ( x ) dx = ± b -∞ f ( x ) dx = ± -∞ f ( x ) dx = Remark 1.1. ± -∞ f ( x ) dx is divergent iF at least one oF the ± a f ( x ) dx or ± a -∞ f ( x ) dx is divergent. Example 1.1. ² 1 1 x dx Example 1.2. ² 0 -∞ xe x dx Example 1.3. ² -∞ x x 2 +1 dx 1
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Section 7.8 Improper Integrals 2 2. The form 1 x p Theorem 2.1. (1) If p> 1 the ± 1 dx x p is (2) If p 1 the ± 1 dx x p is 3. Discontinuous Integrand Example 3.1. ² 1 0 1 3 x 4 dx Example 3.2. ² 3 - 2 1 x 4 dx Example 3.3. ² 1
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Unformatted text preview: ln x dx Section 7.8 Improper Integrals 3 4. Comparison Theorem Theorem 4.1. Suppose ≤ g ( x ) ≤ f ( x ) for all x in the interval ( a, b ) , a and/or b may be inFnite. (1) If ± b a f ( x ) dx is , then ± b a g ( x ) dx is (2) If ± b a g ( x ) dx is , then ± b a f ( x ) dx is Example 4.1. Is the integral convergent or divergent? ² ∞ e-x 2 dx Example 4.2. Is the integral convergent or divergent? ² ∞ 1 1 √ 1 + x 3 dx Example 4.3. Is the integral convergent or divergent? ² ∞ 1 x + 1 √ x 4-x dx...
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This note was uploaded on 03/14/2011 for the course MAC 2312 taught by Professor Zhang during the Fall '07 term at FSU.

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sec7_8 - ln x dx Section 7.8 Improper Integrals 3 4....

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