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Mathematics Practice Exam

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1. Please evaluate the integrals below. 5 (a) ∫2 1 𝑑𝑥 (𝑥−3)2 5 ∫ 2 1 𝑑𝑥 (𝑥 − 3)2 1 The expression (𝑥−3)2 is undefined at 𝑥 = 3: 3 =∫ 2 5 1 1 𝑑𝑥 + ∫ 2 2 𝑑𝑥 (𝑥 − 3) 3 (𝑥 − 3) Find the limit as t approaches 3− and 3+ : 𝑡 = lim− ∫ 𝑡→3 2 5 1 1 𝑑𝑥 + lim ∫ 𝑑𝑥 2 + 𝑡→3 𝑡 (𝑥 − 3)2 (𝑥 − 3) Calculate the integrals by using power rule where exponent is equal to -2. = lim− [− 𝑡→3 1 1 5 𝑡 ] + lim+ [− ] 𝑥 − 3 2 𝑡→3 𝑥−3 𝑡 Calculate the integral boundaries: = lim− [− 𝑡→3 1 1 1 − 1] + lim+ [− + ] 𝑡→3 𝑡−3 2 𝑡−3 5 ∫ 2 1 𝑑𝑥 = ∞ (𝑥 − 3)2 0 (b) ∫−∞ 𝑥𝑒 2+3𝑥 𝑑𝑥 0 ∫ 𝑥𝑒 2+3𝑥 𝑑𝑥 −∞ By using integration by parts, compute indefinite integral first: let 𝑢 = 𝑥 and 𝑣 ′ = 𝑒 2+3𝑥 𝑢′ = 1 1 𝑣 = ∫ 𝑒 2+3𝑥 = 𝑒 2+3𝑥 3 1 1 𝑢𝑣 − ∫ 𝑢′ 𝑣 = 𝑥 ( 𝑒 2+3𝑥 ) − ∫ 𝑒 2+3𝑥 𝑑𝑥 3 3 1 1 = 𝑒 2+3𝑥 𝑥 − 𝑒 2+3𝑥 3 9 Calculate the integral boundaries by calculating limit as x approaches −∞ and 0. 1 2+3𝑥 1 1 1 𝑒 𝑥 − 𝑒 2+3𝑥 ) − ( lim− 𝑒 2+3𝑥 𝑥 − 𝑒 2+3𝑥 ) 𝑥→−∞ 3 𝑥→0 3 9 9 = ( lim 1 = − 𝑒2 − 0 9 0 1 ∫ 𝑥𝑒 2+3𝑥 𝑑𝑥 = − 𝑒 2 9 −∞ ∞ 2. Use the comparison theorem to determine whether the integral ∫2 ∞1 If ∫1 𝑥 ...
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