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11Breview8sol - econ 11b ucsc ams 11b Review Questions 8...

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Unformatted text preview: econ 11b ucsc ams 11b Review Questions 8 Solutions 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 . Use the formula Z du a + be ku = ku- ln a + be ku ak + C , (with a = 1, b = 0 . 25 and k =- . 1). f. Z 4(ln x ) 2 3 x √ 2 + 7 ln x dx = 4 3 Z (ln x ) 2 x √ 2 + 7 ln x dx = 4 3 Z u 2 du √ 2 + 7 u , (substituting u = ln x , du = 1 x dx ), = 4 3 2(147 u 2- 56 u + 32) √ 2 + 7 u 5145 + C = 8(147(ln x ) 2- 56 ln x + 32) √ 2 + 7 ln x 15435 + C ....
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This note was uploaded on 09/16/2011 for the course ECON 11B taught by Professor Binici during the Spring '08 term at UCSC.

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11Breview8sol - econ 11b ucsc ams 11b Review Questions 8...

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