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NAMES: MATH 152 March 4, 2015 QUIZ 5 • Show all your work and indicate your final answer clearly. You will be graded not merely on the final answer, but also on the work leading up to it. 1. (3 points) Find the partial fraction decomposition (including coefficients) of x−8 2 x − 7x + 10 x−8 A B x−8 + x−2 . Then Solution: Let x2 −7x+10 = (x−5)(x−2) = x−5 x − 8 = A(x − 2) + B(x − 5). When x = 2, −3B = −6 so B = 2. When x = 5, 3A = −3 so A = −1. Thus A = −1, B = 2 2. (3 points) Evaluate the integral. If the integral does not converge, write ”diverges”. Z ∞ e−2x dx 0 Solution: By definition, Z ∞ e 0 −2x Z t e−2x dx 0 t 1 −2x = lim − e t→∞ 2 0 1 −2t 1 0 = lim − e + e t→∞ 2 2 1 −2t 1 = lim − e + t→∞ 2 2 1 = . 2 dx = lim t→∞ NAMES: MATH 152 March 4, 2015 3. (3 points) Evaluate the integral x2 dx. x+3 Solution: Since the degree of the numerator is higher than the degree of the denominator, x2 9 use long division to find that x+3 = x − 3 + x+3 so Z Z x2 9 x2 dx = x − 3 + dx = − 3x + 9 ln |x + 3| + C x+3 x+3 2 Z