11Brev2andahalfsol

11Brev2andahalfsol - ucsc econ/ams 11b Review Questions 2.5...

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Unformatted text preview: ucsc econ/ams 11b Review Questions 2.5 Table of integrals and separable differential equations 1. Compute the following integrals a. Z 5 x 7- 2 xdx = 5 Z x 7- 2 xdx = 10(- 6 x- 14)(7- 2 x ) 3 / 2 60 + C =- (3 x + 7)(7- 2 x ) 3 / 2 3 + C . Use the formula Z u a + budu = 2(3 bu- 2 a )( a + bu ) 3 / 2 15 b 2 + C , with a = 7 and b =- 2. b. Z 7 t 2 + 3 t- 1 2 + 5 t dt = 7 Z t 2 dt 2 + 5 t + 3 Z tdt 2 + 5 t- Z dt 2 + 5 t = 7 t 2 10- 2 t 25 + 4 125 ln | 2 + 5 t | +3 t 5- 2 25 ln | 2 + 5 t | - 1 5 ln | 2 + 5 t | + C = 7 10 t 2 + 1 25 t- 27 125 ln | 2 + 5 t | + C . Use the formulas (i) Z du a + bu = 1 b ln | a + bu | + C, (ii) Z udu a + bu = u b- a b 2 ln | a + bu | + C, and (iii) Z u 2 du a + bu = u 2 2 b- au b 2 + a 2 b 3 ln | a + bu | + C , ( all with a = 2 and b = 5). c. Z 500 t 2 e- . 04 t dt = 500 t 2 e- . 04 t- . 04 + 2 . 04 Z te- . 04 t dt = 500 t 2 e- . 04 t- . 04 + 2 . 04 e- . 04 t . 0016 (- . 04 t- 1) + C =- 12500 e- . 04 t ( t 2 + 50 t + 1250 ) + C . Use the formulas (i) Z u n e au du = u n e au a- n a Z u n- 1 e au du , (with n = 2 and a =- . 04), and (ii) Z ue au du = e au a 2 ( au- 1) + C , (with a =- . 04). d. Z 3 e 2 x 4 + e x dx = 3 Z e x 4 + e x e x dx = 3 Z u 4 + u du , (substituting u = e x and du = e x dx ), = 3 2( u- 8) 4 + u 3 + C = 2( e x- 8) 4 + e x + C . Use the formula Z udu a + bu = 2( bu- 2 a ) a + bu 3 b 2 + C , (with a = 4 and b = 1). e. Z 300 1 + 0 . 25 e- . 1 t dt = 300- . 01 t- ln | 1 + 0 . 25 e- . 01 t |- . 01 + C = 30000 ( . 01 t + ln | 1 + 0 . 25 e- . 01 t | ) + C ....
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11Brev2andahalfsol - ucsc econ/ams 11b Review Questions 2.5...

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