Question

find the integral of ∫(((2x+7)))/(x^3-3x^2+2x)

Answer 1

Find the integral: $\int \frac{2x + 7}{x^3 - 3x^2 + 2x} dx$.

**Solution:**

$$\begin{aligned} & \frac{2x + 7}{x^3 - 3x^2 + 2x} = \frac{2x + 7}{x(x^2 - 3x + 2)} = \frac{2x + 7}{x(x - 2)(x - 1)} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x - 1} \\ & A = \frac{2x + 7}{(x^2 - 3x + 2)} \Bigg|_x = 0 = \frac{7}{2} \\ & B = \frac{2x + 7}{x(x - 1)} \Bigg|_x = 2 = \frac{11}{2} \\ & C = \frac{2x + 7}{x(x - 2)} \Bigg|_x = 1 = -9 \\ & \int \frac{2x + 7}{x^3 - 3x^2 + 2x} dx \\ & \quad = \int \left( \frac{7}{2} \frac{1}{x} + \frac{11}{2} \frac{1}{x - 2} - \frac{9}{x - 1} \right) dx = \frac{7}{2} \ln |x| + \frac{11}{2} \ln |x - 2| \\ & \quad - 9 \ln |x - 1| + C, \end{aligned}$$

where $C = \text{const}$.

**Answer:** $\int \frac{2x + 7}{x^3 - 3x^2 + 2x} dx = \frac{7}{2} \ln |x| + \frac{11}{2} \ln |x - 2| - 9 \ln |x - 1| + C$.