Please Answer all the questions. make sure the handwriting are same for all the answers. thanks

b. (6 marks) Assignment 3 (100 marks total) sin 2 t cos ' t dt 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 answer a. (6 marks) a. (6 marks) Vxinx dx x/9 - 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: 2. [8 marks] a. (6 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 x"edx = x"ex - n fx"-le*dx. 4x3 + x+ 2 b. (6 marks) x2 (x2 + 1) -dx Use the reduction formula in a. to find S x3e*dx. 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, n/2]. Find the volume of the solid obtained by rotating R around the y axis. Levedx 4. [12 marks] b. (6 marks) Evaluate: dx x2 V4x2 - 1 a. (6 marks) tan ? x sec * x dx8. [8 marks] Use: a. (4 marks) The Trapezoidal Rule b. (4 marks) Simpson's Rule to approximate f0? 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) I x b. (4 marks) 3 1 f dx 2 3x