MATH 1241: Calculus II A3-1 Assignment 3 (100 marks total) Note: Give exact answers where possible (for example, it is exact but 3.142 is approximate). 1. [18 marks] Evaluate: a. (6 marks) x ln x dx b. (6 marks) x sec x dx c. (6 marks) tarctan(t) dt 2. [8 marks] a. (2 marks) Use integration by parts to verify the validity of the following reduction formula: x"exdx=x"e*-nfx"-e* dx. b. (6 marks) Use the reduction formula in a. to find fxe*dx. 3. [10 marks] Let R be the region between the curve y = cos x and the x axis on the interval [0,/2]. Find the volume of the solid obtained by rotating R around the y axis. 4. [12 marks] Evaluate: a. (6 marks) tanx sec x dx A3-2 b. (6 marks) S sint cos t dt Assignment 3 5. [12 marks] Evaluate the following by using an appropriate trigonometric substitution. Give exact answers a. (6 marks) 3/4 x9-4x dx b. (6 marks) x (25+ x)5/2 dx 6. [12 marks] Evaluate: a. (6 marks) x-6x-12 x3-4x b. (6 marks) 4x +x+2 x(x + 1) dx -dx 7. [12 marks] Evaluate. a. (6 marks) b. (6 marks) dx MATH 1241: Calculus II A3-3 8. [8 marks] Use: a. (4 marks) The Trapezoidal Rule b. (4 marks) Simpson's Rule to approximate 1 + cos x dx with n = 8. Round your answer to six decimal places. 9. [8 marks] Determine whether each integral is convergent or divergent. Evaluate any that is convergent. a. (4 marks) 00 x -dx 10 x +9 b. (4 marks) 1 dx