MATH 1241: Calculus II A3-1 A3-2 Assignment 3 b. (6 marks) Assignment 3 (100 marks total) sin 2 t cost at Note: Give exact answers where possible (for example, It is exact but 3.142 is approximate). 5. [12 marks] 1. [18 marks] Evaluate the following by using an appropriate trigonometric substitution. Give Evaluate: exact answers a. (6 marks) a. (6 marks) Vx Inx dx x9 - 4x2 dx b. (6 marks) b. (6 marks) x sec2 x dx x2 (25 + x2 ) 5/2 dx c. (6 marks) t arctan(t) dt 6. [12 marks] Evaluate: a. (6 marks) 2. [8 marks] x4 - 6x - 12 a. (2 marks) x3 - 4x - dx Use integration by parts to verify the validity of the following reduction b. (6 marks) formula: S xe*dx = xex -nj x-lexdx. 4x3 + x+2 b. (6 marks) x2 ( x2 + 1 ) -dx Use the reduction formula in a. to find S x3exdx. 7. [12 marks] 3. [10 marks] Evaluate. Let R be the region between the curve y = cos x and the x axis on the interval a. (6 marks) [0, It/2]. Find the volume of the solid obtained by rotating R around the y axis. Levi dx 4. [12 marks] b. (6 marks) Evaluate: dx x2 V 4x2 - 1 a. (6 marks) tan ? x sec* x dx TRU Open Learning TRU Open LearningMATH 1241: Calculus II A3-3 8. [8 marks] Use: a. (4 marks) The Trapezoidal Rule b. (4 marks) Simpson's Rule to approximate S2 V1 + 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) 72 4 9 dx b. (4 marks) - x dx Select resources associated with the following course textbook are used with permission from Cengage Learning: Stewart, J. (2016). Single variable calculus: Early transcendentals (8th ed.). Boston, MA: Cengage Learning. TRU Open Learning