MA132: Calculus II Name:Exam 1 (14 February 2002) Student Number:(10) Problem 1. Find the integral:Zx cos(3x) dx.(8) Problem 2. Find the integral:Z[ln(x)]3xdx.(8) Problem 3. Find the integral:Zsin4(x) cos3(x) dx.(12) Problem 4. Find the integral:Z√x2− 9xdx.(12) Problem 5. Find the integral:Z2x3(x + 1)(x2+ 1)dx.(10) Problem 6. Find the value:Z∞0x3e−x4dx.(10) Problem 7. Snow falls at the following rate (in cm/hr) one day in Buffalo, New York:1 p.m. 3 p.m. 5 p.m. 7 p.m. 9 p.m.1 cm/hr 5 cm/hr 3 cm/hr 2 cm/hr 1 cm/hrEstimate the total snowfall (in cm) from 1 p.m. to 9 p.m.(a) Using the midpoint rule with n = 2 subintervals.(b) Using the trapezoidal rule with n = 4 subintervals.(8) Problem 8. For each of the following, give the form of the partial fraction decomposition. Donot solve for the unknown constants.(a)2x4− 5x3+ 2x2− 7x + 8(x + 3)2(x − 1)(x − 4)3(b)x3− 4x + 9(x2+ 2)2(x2− 4x + 5)(5) Problem 9. (Multiple choice) Find the appropriate trigonometric substitution for the integralZ√5 + 4x − x2dx(Do not integrate—just find the substitution.)(a) x = 2 + 3 sin(θ )(b) x = 2 + 3 tan(θ)(c) x = 2 + 3 sec(θ)(d) x = −2 + 3 sin(θ )(e) x = −2 + 3 tan(θ)(f) x = −2 + 3 sec(θ )(5) Problem 10. (Multiple choice) Determine the limit: limx→01 − e2xsin(3x)(a) undefined(b)23(c) −23(d)32(e) −32(f) 1(5) Problem 11. (Multiple choice) Determine the limit: limx→0x2cos(x)(a) undefined(b) 0(c) −1(d) 1(e) −2(f) 2(5) Problem 12. (Multiple choice) Determine the limit: limx→0(1 − 3x)2/x(a) 1(b) e−3(c) e3(d) e2(e) e6(f)