Midterm-Exam Aid

Finally if has a discontinuity at where and both

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Unformatted text preview: ￿￿ ￿ ￿ ￿￿￿ ￿￿￿ ￿￿ ￿￿ ￿ ￿￿￿ ￿￿￿ ￿￿ ￿ ￿ ￿￿￿ ￿￿￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿ ￿￿ ￿￿￿ ￿￿￿ ￿￿ ￿￿ ￿ ￿ ￿￿ ￿￿￿ ￿￿ ￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿￿ ￿ ￿ ￿￿ E XAMPLE 3. Calculate ￿ ￿￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿￿ S OLUTION. We begin by simplifying the integrand. The similarity of the sine factors leads us to use the identity ￿ ￿￿￿￿￿￿ ￿￿￿￿￿ ￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿ ￿￿￿ ￿ Then ￿ ￿ ￿ ￿￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿ ￿￿￿￿￿￿￿￿ ￿ ￿ ￿￿￿￿￿￿￿￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿ ￿￿￿￿￿￿￿ ￿￿￿ ￿ ￿ The first integral can be completed by inspection or by using the substitution ￿ ￿ ￿￿ : ￿ ￿ ￿￿ ￿￿￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿￿￿￿ ￿ ￿ ￿￿ ￿ The second integral is a special case of Example 2, with ￿ ￿ ￿ and ￿ ￿ ￿: ￿ ￿￿ ￿￿ ￿￿￿￿￿￿￿ ￿￿ ￿ ￿￿ ￿￿￿￿￿￿￿ ￿ ￿￿￿￿￿￿￿￿ ￿ ￿￿ ￿￿￿ Together, ￿ ￿￿ ￿ ￿￿ ￿￿￿￿￿￿￿ ￿ ￿ ￿￿ ￿￿￿￿￿￿￿ ￿ ￿￿￿￿￿￿￿￿ ￿ ￿ ￿￿ ￿￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿￿￿￿ ￿ ￿ ￿￿ ￿￿￿￿￿￿￿ ￿ ￿￿￿￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿ ￿￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿ ￿ E XAMPLE 4. Calculate ￿ ￿￿ ￿￿ ￿￿ ￿￿￿￿￿ ￿ ￿￿￿ ￿￿ ￿ ￿ ￿￿￿ ￿￿ S OLUTION. We have a radical, we we try the substitution ￿ ￿ ￿ ￿ ￿. Then ￿ ￿ ￿￿￿ ￿ ￿￿￿ , so that ￿ 4 . To alter the bounds, when ￿ ￿ ￿￿, then ￿ ￿ ￿￿￿￿ ￿ ￿ ￿ ￿ and ￿￿ ￿ ￿￿￿￿ ￿ ￿￿￿￿￿￿ ￿￿, and ￿ ￿ ￿ ￿￿￿ ￿ ￿￿ ￿￿ when ￿ ￿ ￿￿, then ￿ ￿ ￿￿ ￿ ￿ ￿ ￿. We thus have ￿ ￿￿ ￿￿ ￿￿ ￿￿￿￿￿ ￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿ ￿ ￿￿￿ ￿￿ ￿￿￿ ￿ ￿￿￿ ￿￿ ￿￿ ￿ ￿ ￿￿￿ ￿￿ ￿ ￿￿ ￿￿￿ ￿￿ ￿￿￿ ￿￿ ￿￿￿ ￿ ￿￿ ￿ 9 WATERLOO SOS E XAM -AID: MATH138 M IDTERM Since we have a rational function, we can try to apply partial fractions. Write ￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿￿ which, after multiplying by ￿￿ ￿￿￿ ￿ ￿￿, yields ￿ ￿ ￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿ ￿ ￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿￿ ￿￿ ￿ ￿￿￿ ￿￿ Comparing coefficients gives ￿ ￿ ￿ ￿ ￿￿ Solving, ￿￿ Then, ￿ ￿￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿￿￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿￿￿ ￿ ￿￿￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿￿￿￿￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿￿ ￿￿ ￿ ￿￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿￿ ￿￿￿ ￿￿￿ ￿￿￿ ￿￿ ￿￿ ￿￿ ￿￿ ￿￿ ￿￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿￿￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿￿￿￿￿ ￿ ￿￿￿￿￿￿￿￿￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿￿￿￿￿ ￿ ￿ ￿￿￿￿￿￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿￿￿ ￿ ￿ ￿...
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This document was uploaded on 03/04/2014 for the course MATH 138 at Waterloo.

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