Answer:
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Explanation:
To judge the elemental ∫(3x / (4x^2 - 3x + 2)) dx, we can use the arrangement of biased parts.First, allow's determinant the common factor:4x^2 - 3x + 2 = (4x - 1)(x - 2)Now we can express the integrand as a total of biased parts:(3x / (4x^2 - 3x + 2)) = A / (4x - 1) + B / (x - 2)To find the principles of A and B, we need to decide the numerators of the prejudiced parts. We be able this by cross-duplication:3x = A(x - 2) + B(4x - 1)Expanding the kindliness the equating, we receive:3x = Ax - 2A + 4Bx - BGrouping the agreements accompanying x and the perpetual agreements, we have:3x = (A + 4B)x + (-2A - B)To balance the coefficients of x and the uninterrupted agreements, we catch the following whole of equatings:A + 4B = 3-2A - B = 0Solving this whole of equatings, we find A = 2/7 and B = 9/7.Now we can revise the original complete utilizing the prejudiced parts:∫(3x / (4x^2 - 3x + 2)) dx = ∫(2/7) / (4x - 1) dx + ∫(9/7) / (x - 2) dxIntegrating each term individually, we have:∫(2/7) / (4x - 1) dx = (2/7) * (1/4) * ln|4x - 1| + C1∫(9/7) / (x - 2) dx = (9/7) * ln|x - 2| + C2Where C1 and C2 are unification whole.Therefore, the resolution to the basic is:∫(3x / (4x^2 - 3x + 2)) dx = (2/7) * (1/4) * ln|4x - 1| + (9/7) * ln|x - 2| + CWhere C = C1 + C2 is the last unification loyal.