This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for the integrals using partial fractions: 3.1 (x^3 + 5x^2 - 3x - 1)/((3x+2)^2(x^2-1)) dx Step 1: Factor the denominator completely and set up the partial fraction decomposition. The denominator is (3x+2)^2(x^2-1) = (3x+2)^2(x-1)(x+1). The partial fraction decomposition is: (x^3 + 5x^2 - 3x - 1)/((3x+2)^2(x-1)(x+1)) = (A)/(3x+2) + (B)/((3x+2)^2) + (C)/(x-1) + (D)/(x+1) Multiply both sides by the common denominator (3x+2)^2(x-1)(x+1): x^3 + 5x^2 - 3x - 1 = A(3x+2)(x-1)(x+1) + B(x-1)(x+1) + C(3x+2)^2(x+1) + D(3x+2)^2(x-1) Step 2: Solve for the coefficients A, B, C, and D. • Set x = 1: 1^3 + 5(1)^2 - 3(1) - 1 = C(3(1)+2)^2(1+1) 2 = C(5)^2(2) 2 = 50C C = (2)/(50) = (1)/(25) • Set x = -1: (-1)^3 + 5(-1)^2 - 3(-1) - 1 = D(3(-1)+2)^2(-1-1) -1 + 5 + 3 - 1 = D(-1)^2(-2) 6 = -2D D = -3 • Set x = -(2)/(3): (-(2)/(3))^3 + 5(-(2)/(3))^2 - 3(-(2)/(3)) - 1 = B(-(2)/(3)-1)(-(2)/(3)+1) -(8)/(27) + (20)/(9) + 2 - 1 = B(-(5)/(3))((1)/(3)) (-8 + 60 + 27)/(27) = -(5)/(9)B (79)/(27) = -(5)/(9)B B = (79)/(27) · (-(9)/(5)) = -(79)/(15) • Compare coefficients of x^3: The expanded equation is x^3 + 5x^2 - 3x - 1 = A(3x^3 + 2x^2 - 3x - 2) + B(x^2-1) + C(9x^3+21x^2+16x+4) + D(9x^3+3x^2-8x-4). Comparing coefficients of x^3: 1 = 3A + 9C + 9D 1 = 3A + 9((1)/(25)) + 9(-3) 1 = 3A + (9)/(25) - 27 1 = 3A + (9 - 675)/(25) 1 = 3A - (666)/(25) 3A = 1 + (666)/(25) = (25 + 666)/(25) = (691)/(25) A = (691)/(75) Step 3: Integrate each term of the partial fraction decomposition. ( (691/75)/(3x+2) + (-79/15)/((3x+2)^2) + (1/25)/(x-1) + (-3)/(x+1) ) dx = (691)/(75) (1)/(3x+2) dx - (79)/(15) (3x+2)^-2 dx + (1)/(25) (1)/(x-1) dx - 3 (1)/(x+1) dx = (691)/(75) ((1)/(3)|3x+2|) - (79)/(15) ((1)/(3)(3x+2)^-1-1) + (1)/(25)|x-1| - 3|x+1| + K Step 4: Simplify and combine the results. (691)/(225)|3x+2| + (79)/(45(3x+2)) + (1)/(25)|x-1| - 3|x+1| + K 3.2 (8x^3 - x^2 + 4x - 3)/((3-2x)^3) dx Step 1: Use a substitution to simplify the denominator. Let u = 3-2x. Then 2x = 3-u, so x = (3-u)/(2). Also, du = -2 dx, which means dx = -(1)/(2) du. Step 2: Substitute x and dx into the integral and simplify the numerator. Numerator: 8x^3 - x^2 + 4x - 3 = 8((3-u)/(2))^3 - ((3-u)/(2))^2 + 4((3-u)/(2)) - 3 = 8((3-u)^3)/(8) - ((3-u)^2)/(4) + 2(3-u) - 3 = (27 - 27u + 9u^2 - u^3) - (1)/(4)(9 - 6u + u^2) + (6 - 2u) - 3 = -u^3 + (9 - (1)/(4))u^2 + (-27 + (3)/(2) - 2)u + (27 - (9)/(4) + 6 - 3) = -u^3 + (35)/(4)u^2 - (55)/(2)u + (111)/(4) The integral becomes: (-u^3 + 35)/(4)u^2 - (55)/(2)u + (111)/(4)u^3 (-(1)/(2)) du = -(1)/(2) ( (-u^3)/(u^3) + (35u^2)/(4u^3) - (55u)/(2u^3) + (111)/(4u^3) ) du = -(1)/(2) ( -1 + (35)/(4u) - (55)/(2u^2) + (111)/(4u^3) ) du Step 3: Integrate each term with respect to u. = -(1)/(2) ( -u + (35)/(4)|u| - (55)/(2)(-(1)/(u)) + (111)/(4)(-(1)/(2u^2)) ) + C = -(1)/(2) ( -u + (35)/(4)|u| + (55)/(2u) - (111)/(8u^2) ) + C = (1)/(2)u - (35)/(8)|u| - (55)/(4u) + (111)/(16u^2) + C Step 4: Substitute back u = 3-2x. (1)/(2)(3-2x) - (35)/(8)|3-2x| - (55)/(4(3-2x)) + (111)/(16(3-2x)^2) + C The constant term (3)/(2) from (1)/(2)(3-2x) can be absorbed into the integration constant C. So, an equivalent form is: -x - (35)/(8)|3-2x| - (55)/(4(3-2x)) + (111)/(16(3-2x)^2) + C 3 done, 2 left today. You're making progress.