MA132 Name:Exam 1 Student Number:2 June 2006Instructions:• Show your work.• Answers without sufficient justification may not receive full cr edit.• No books, notes, or calculators.• Time limit: 90 minutes.• Some integrals which might help:Zsec(x) dx = ln |sec(x) + tan(x)| + CZcsc(x) dx = ln |csc(x) − cot(x)| + CDo not write below this lineProblem 1: 12 pointsProblem 2: 8 pointsProblem 3: 12 pointsProblem 4: 8 pointsProblem 5: 10 pointsProblem 6: 10 pointsProblem 7: 8 pointsProblem 8: 12 pointsProblem 9: 12 pointsProblem 10: 8 pointsScore = 100 points total(12) Problem 1. Evaluate the integral:Ztan2(x) sec4(x) dxPage 2 of 11(8) Problem 2. For each of the following, give the form of the partial fraction decomposition.Do not solve for the unknown constants. Hint: Factor where possible.(a)x3+ 3x2− 5(x − 2)(x + 5)2(x2+ 3)2(b)x4− 2x2+ 17x3(x2− 1)(x2+ x − 6)Page 3 of 11(12) Problem 3. Evaluate the integral:Z1√x2− 2x + 5dxPage 4 of 11(8) Problem 4. The speed v(t) of a crawling slug1for certain values of time t is given in the table:t (minutes) 0 2 4 6 8v(t) (cm/minute) 2 4 5 3 1Estimate the distance traveled by the s lug from t = 0 to t = 8 minutes.(a) Use the Trapezoidal rule with n = 4 subintervals.(b) Use the Midpoint rule with n = 2 subintervals.1measured with a radar gunPage 5 of 11(10) Problem 5. Evaluate the integral:Zxx + 3dxPage 6 of 11(10) Problem 6. Evaluate the integral:Zθ2sin(5θ) dθPage 7 of 11(8) Problem 7. Evaluate the following limits. If you use L’Hospital’s Rule, show where you us ed itand that it applies.(a) limx→∞x sin1x(b) limx→∞x2/xPage 8 of 11(12) Problem 8. Evaluate the integral:Zx − 6x3− 2x2dxPage 9 of 11(12) Problem 9. Determine whether the following integral is convergent or divergent. If it is conver-gent, evaluate it.Z∞11(2x + 1)2dxPage 10 of 11(8) Problem 10. The integralZ2−1f(x) dx of the function f (x) shown in the graph below is to beestimated by numerical integration. Rank the following five numbers in order of size f rom smallestto largest:• Ln: left endpoint (left sum) approximation• Rn: right endpoint (right sum) approximation• Mn: midpoint approximation• Tn: trapezoidal approximation• I: the exact value of the integralZ2−1f(x) dxNote: each approximation is computed using the same number n of subintervals.-x6f(x)−10 1 2Page 11 of