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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Step 1: Perform polynomial long division for integral 3.2. For integral 3.2, the degree of the numerator (4x^3 - 6x^2 + 5x - 4) is 3, and the degree of the denominator (2x^2 + x - 6) is 2. Since the numerator's degree is greater than or equal to the denominator's degree, we must perform polynomial long division first. c|cc cc 2r2x & -4 \\ 2-5 2x^2+x-6 & 4x^3 & -6x^2 & +5x & -4 \\ 2r-(4x^3 & +2x^2 & -12x) \\ 2-4 2r0 & -8x^2 & +17x & -4 \\ 2r & -(-8x^2 & -4x & +24) \\ 3-5 2r & 0 & 21x & -28 \\ So, (4x^3 - 6x^2 + 5x - 4)/(2x^2 + x - 6) = 2x - 4 + (21x - 28)/(2x^2 + x - 6). Step 2: Factor the denominator for integral 3.2. The denominator is 2x^2 + x - 6. We find the roots using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a: x = -1 ± sqrt(1^2 - 4(2)(-6))2(2) = -1 ± sqrt(1 + 48)4 = -1 ± sqrt(49)4 = (-1 ± 7)/(4) The roots are x_1 = (-1+7)/(4) = (6)/(4) = (3)/(2) and x_2 = (-1-7)/(4) = (-8)/(4) = -2. So, 2x^2 + x - 6 = 2(x - (3)/(2))(x + 2) = (2x - 3)(x + 2). Step 3: Decompose the remainder term into partial fractions for integral 3.2. We set up the partial fraction decomposition for (21x - 28)/((2x - 3)(x + 2)): (21x - 28)/((2x - 3)(x + 2)) = (A)/(2x - 3) + (B)/(x + 2) Multiply both sides by (2x - 3)(x + 2): 21x - 28 = A(x + 2) + B(2x - 3) To find A, set x = (3)/(2): 21((3)/(2)) - 28 = A((3)/(2) + 2) + B(0) (63)/(2) - (56)/(2) = A((7)/(2)) (7)/(2) = A