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RandolphT2W98

# RandolphT2W98 - improper integral converges or diverges If...

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Winter 1992 Math 21 Test #2 Name 1. (10 points each) Find only FIVE of the following six indefinite integrals. Show each step. Circle the integral for problem that you DO NOT want to be graded . ( This will be graded on an all-or-nothing basis for 4 points extra credit.) (a) Z cos(3 x ) cos(4 x ) dx (b) Z x - 3 x 2 + 1 dx (c) Z 1 4 x 2 + 1 dx (d) Z tan 3 x dx (e) Z x 2 ln x dx (f) Z p 9 - x 2 dx 2. (10) Use partial fractions to express the integral as a sum of simpler integrals. Z 5 x 2 + 3 x - 2 x 2 ( x + 2) dx DO NOT EVALUATE the integral. 3. (a) (4 pts) Use the trapezoid rule with n = 4 to approximate Z 1 0 e - x 2 dx . (b) (6 pts) If T 4 is the number you found in (a), how large is the difference: fl fl fl fl Z 1 0 e - x 2 dx - T 4 fl fl fl fl ? You may use the fact that for f ( x ) = e - x 2 , f 00 ( x ) 2 for x [0 , 1]. (c) (6 pts) How large should n be in order for the trapezoid rule to produce a value, T n , that approxi- mates R 1 0 e - x 2 dx to within 0.001? 4. (6 pts) A definite integral has been calculated for you below. Determine whether the corresponding
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Unformatted text preview: improper integral converges or diverges . If it converges, specify its value. Z t 1 ln x x 2 dx = ( using “parts” ) =-ln t t-1 t + 1. Therefore, Z ∞ 1 ln x x 2 dx = 5. (10) Show all your work in determining the value (if it exists) of the following improper integral. If it diverges show your work in determining this. Z 3 2 3 √ x-2 dx 6. (8) Use the comparison thoerem, explaining how it applies, to determine whether the integral is convergent or divergent . You DO NOT need to evaluate the integral. (You may use, without proof, any knowledge about convergence or divergence of the improper integrals involving 1 /x , 1 /x 2 , e-x , etc., but be speciﬁc.) Z ∞ 1 sin 2 x x 2 dx...
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