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Mathematics Practice Exam

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1. Please evaluate the integrals below. 5 (a) ∫2 1 𝑑π‘₯ (π‘₯βˆ’3)2 5 ∫ 2 1 𝑑π‘₯ (π‘₯ βˆ’ 3)2 1 The expression (π‘₯βˆ’3)2 is undefined at π‘₯ = 3: 3 =∫ 2 5 1 1 𝑑π‘₯ + ∫ 2 2 𝑑π‘₯ (π‘₯ βˆ’ 3) 3 (π‘₯ βˆ’ 3) Find the limit as t approaches 3βˆ’ and 3+ : 𝑑 = limβˆ’ ∫ 𝑑→3 2 5 1 1 𝑑π‘₯ + lim ∫ 𝑑π‘₯ 2 + 𝑑→3 𝑑 (π‘₯ βˆ’ 3)2 (π‘₯ βˆ’ 3) Calculate the integrals by using power rule where exponent is equal to -2. = limβˆ’ [βˆ’ 𝑑→3 1 1 5 𝑑 ] + lim+ [βˆ’ ] π‘₯ βˆ’ 3 2 𝑑→3 π‘₯βˆ’3 𝑑 Calculate the integral boundaries: = limβˆ’ [βˆ’ 𝑑→3 1 1 1 βˆ’ 1] + lim+ [βˆ’ + ] 𝑑→3 π‘‘βˆ’3 2 π‘‘βˆ’3 5 ∫ 2 1 𝑑π‘₯ = ∞ (π‘₯ βˆ’ 3)2 0 (b) βˆ«βˆ’βˆž π‘₯𝑒 2+3π‘₯ 𝑑π‘₯ 0 ∫ π‘₯𝑒 2+3π‘₯ 𝑑π‘₯ βˆ’βˆž By using integration by parts, compute indefinite integral first: let 𝑒 = π‘₯ and 𝑣 β€² = 𝑒 2+3π‘₯ 𝑒′ = 1 1 𝑣 = ∫ 𝑒 2+3π‘₯ = 𝑒 2+3π‘₯ 3 1 1 𝑒𝑣 βˆ’ ∫ 𝑒′ 𝑣 = π‘₯ ( 𝑒 2+3π‘₯ ) βˆ’ ∫ 𝑒 2+3π‘₯ 𝑑π‘₯ 3 3 1 1 = 𝑒 2+3π‘₯ π‘₯ βˆ’ 𝑒 2+3π‘₯ 3 9 Calculate the integral boundaries by calculating limit as x approaches βˆ’βˆž and 0. 1 2+3π‘₯ 1 1 1 𝑒 π‘₯ βˆ’ 𝑒 2+3π‘₯ ) βˆ’ ( limβˆ’ 𝑒 2+3π‘₯ π‘₯ βˆ’ 𝑒 2+3π‘₯ ) π‘₯β†’βˆ’βˆž 3 π‘₯β†’0 3 9 9 = ( lim 1 = βˆ’ 𝑒2 βˆ’ 0 9 0 1 ∫ π‘₯𝑒 2+3π‘₯ 𝑑π‘₯ = βˆ’ 𝑒 2 9 βˆ’βˆž ∞ 2. Use the comparison theorem to determine whether the integral ∫2 ∞1 If ∫1 π‘₯ ...
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