integral of (x^2 - 2x)(x^3 - 3x^2 + 3)^(2/3) dx Thanks! Solution x^3 - 3x^2 + 2x

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Explanation / Answer

x^3 - 3x^2 + 2x => x * (x^2 - 3x + 2) => x * (x - 2) * (x - 1) A/x + B/(x - 2) + C/(x - 1) = (-x^2 - 2x + 2) / (x^3 - 3x^2 + 2x) (A * (x - 1) * (x - 2) + B * x * (x - 1) + C * x * (x - 2)) / (x^3 - 3x^2 + 2x) = (-x^2 - 2x + 2) / (x^3 - 3x^2 + 2x) A * (x^2 - 3x + 2) + B * (x^2 - x) + C * (x^2 - 2x) = -x^2 - 2x + 2 Ax^2 + Bx^2 + Cx^2 - 3Ax - Bx - 2Cx + 2A = -x^2 - 2x + 2 2A = 2 A = 1 Ax^2 + Bx^2 + Cx^2 = -x^2 A + B + C = -1 1 + B + C = -1 B + C = -2 -3Ax - Bx - 2Cx = -2x -3A - B - 2C = -2 3A + B + 2C = 2 3 * 1 + B + 2C = 2 3 + B + 2C = 2 B + 2C = -1 B = -2 - C B = -1 - 2C -2 - C = -1 - 2C 2 + C = 1 + 2C 2 - 1 = 2C - C 1 = C B = -2 - C B = -2 - 1 B = -3 So we have: 1/x - 3/(x - 2) + 1/(x - 1) Integrate with respect to x ln|x| - 3 * ln|x - 2| + ln|x - 1| + C Or we could combine them into a single logarithm: ln|x * (x - 1)| - ln|(x - 2)^3| + C ln|x * (x - 1) / (x - 2)^3| + C

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