MATH 1241: Calculus II Assignment 3 (100 marks total) Note: Give exact answers where possible (for example, è is exact but 3.142 is approximate). 1. [18 marks] Evaluate: a. (6

marks) √ √x In x dx b. (6 marks) [xs c. (6 marks) x sec² x dx S t arctan (t) dt 2. [8 marks] a. (2 marks) Use integration by parts to verify the validity of the following reduction formula: fx¹e* dx = x¹ ex − n f xn-¹ ex dx. b. (6 marks) Use the reduction formula in a. to find f x³exdx. 3. [10 marks] Let R be the region between the curve y = cos x and the x axis on the interval [0, π/2]. Find the volume of the solid obtained by rotating R around the y axis. 4. [12 marks] Evaluate: a. (6 marks) A3-1 S tan² x sec4 x dx TRU Open Learning A3-2 b. (6 marks) S sin² t cos² t dt 5. [12 marks] Evaluate the following by using an appropriate trigonometric substitution. Give exact answers a. (6 marks) 3/4 x√√9-4x² dx 0 b. (6 marks) x² √ (25+x²)5/2 6. [12 marks] Evaluate: a. (6 marks) x² - 6x - 12 .4 x³ - 4x b. (6 marks) 4x³ + x + 2 S² x²(x² + 1) 7. [12 marks] dx Evaluate. a. (6 marks) Sevxdx b. (6 marks) dx x²√4x² - 1 - dx - dx Assignment 3 TRU Open Learning MATH 1241: Calculus II 8. [8 marks] Use: a. (4 marks) The Trapezoidal Rule b. (4 marks) Simpson's Rule TU to approximate ſ √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) S. X x² +9 b. (4 marks) 1 [₁²√3/²= - - dx X A3-3 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