QuestionApril 20, 2026

2. Evaluate the integral int (x^2-6x+17)/(x^3)-4x^(2+x+6)dx

2. Evaluate the integral int (x^2-6x+17)/(x^3)-4x^(2+x+6)dx
2. Evaluate the integral int (x^2-6x+17)/(x^3)-4x^(2+x+6)dx

Solution
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Answer

-3\ln|x-2| + 2\ln|x-3| + 2\ln|x+1| + C Explanation 1. Factor the denominator x^{3} - 4x^{2} + x + 6 → group: (x^{3} - 4x^{2}) + (x + 6) → x^{2}(x - 4) + 1(x + 6) doesn’t match. Try synthetic division with x=2: x^{3}-4x^{2}+x+6 divided by (x-2) → remainder 0. Quotient: x^{2}-2x-3 = (x-3)(x+1). So: (x-2)(x-3)(x+1). 2. Partial fraction decomposition \frac{x^{2}-6x+17}{(x-2)(x-3)(x+1)} = \frac{A}{x-2} + \frac{B}{x-3} + \frac{C}{x+1}. Multiply through: x^{2} - 6x + 17 = A(x-3)(x+1) + B(x-2)(x+1) + C(x-2)(x-3). Substitute: - x=2: 4 - 12 + 17 = A(-1)(3) \Rightarrow 9 = -3A \Rightarrow A = -3. - x=3: 9 - 18 + 17 = B(1)(4) \Rightarrow 8 = 4B \Rightarrow B = 2. - x=-1: 1 + 6 + 17 = C(-3)(-4) \Rightarrow 24 = 12C \Rightarrow C = 2. 3. Integrate each term \int \frac{-3}{x-2} dx + \int \frac{2}{x-3} dx + \int \frac{2}{x+1} dx = -3\ln|x-2| + 2\ln|x-3| + 2\ln|x+1| + C.

Explanation

1. Factor the denominator <br /> $x^{3} - 4x^{2} + x + 6$ → group: $(x^{3} - 4x^{2}) + (x + 6)$ → $x^{2}(x - 4) + 1(x + 6)$ doesn’t match. Try synthetic division with $x=2$: $x^{3}-4x^{2}+x+6$ divided by $(x-2)$ → remainder $0$. Quotient: $x^{2}-2x-3 = (x-3)(x+1)$. So: $(x-2)(x-3)(x+1)$.<br /><br />2. Partial fraction decomposition <br /> $\frac{x^{2}-6x+17}{(x-2)(x-3)(x+1)} = \frac{A}{x-2} + \frac{B}{x-3} + \frac{C}{x+1}$. <br />Multiply through: $x^{2} - 6x + 17 = A(x-3)(x+1) + B(x-2)(x+1) + C(x-2)(x-3)$. <br />Substitute: <br />- $x=2$: $4 - 12 + 17 = A(-1)(3) \Rightarrow 9 = -3A \Rightarrow A = -3$. <br />- $x=3$: $9 - 18 + 17 = B(1)(4) \Rightarrow 8 = 4B \Rightarrow B = 2$. <br />- $x=-1$: $1 + 6 + 17 = C(-3)(-4) \Rightarrow 24 = 12C \Rightarrow C = 2$.<br /><br />3. Integrate each term <br /> $\int \frac{-3}{x-2} dx + \int \frac{2}{x-3} dx + \int \frac{2}{x+1} dx = -3\ln|x-2| + 2\ln|x-3| + 2\ln|x+1| + C$.
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