Math 122 - Spring 2015 - Homework - 6.1 Give complete, well written solutions to the following exercises: 1. Evaluate the following integrals using Integration by Parts. Clearly indicate your choice for u and dv. a. ln x dx. 1 0 b. cos 2e4 d. c. r3 dr. 4 + r2 xe3x dx (6x + 2)2 d. (ln x)2 dx. e. 2. Every dierentiation rule has a corresponding integration rule. a. Use the Quotient Rule for dierentiation and Fundamental Theorem of Calculus to nd a rule to integrate quotients. b. Use your rule to integrate quotients to evaluate sin (x1/2 ) dx x2 and verify your answer is correct. 3. Let Km denote the following indenite integral: Km (x) = xm sin(x) dx for m = 0, 1, 2, . . .. (a) Evaluate K1 (x). (b) Use Integration by Parts to show that Km (x) = xm cos(x) + mxm1 sin(x) m(m 1)Km2 (x). (c) Evaluate K0 (x), K2 (x), and K4 (x). x4 sin(x) dx. (d) Evaluate 0 Math 122 - Spring 2015 - Homework - 6.2 Give complete, well written solutions to the following exercises: 1. Evaluate the following integrals using Trigonometric Substitution. Indicate any identities, substitutions or inverse substitutions used in your calculations. a. b. tan3 x dx. cos3 x 4x2 9 dx. x 1 d. 1 dx. 2 2 0 (1 + x ) 5 + 4x x2 dx. e. e4x e2x + 1 dx. c. 2. Recall the following addition and subtraction identities for the sine function: sin (A + B) = sin A cos B + cos A sin B and sin (A B) = sin A cos B cos A sin B. 1 a. Show that sin A cos B = [sin (A B) + sin (A + B)]. 2 b. Use the result of part a. to evaluate sin 3x cos x dx