Use partial fractions to evaluate \( \int \frac{5 x-13}{x^{2}-5 x+6} d x \). Use "In" to represent the natural log and use "abs()" to indicate absolute value. Fo example, \( \ln \left|u^{3}-8\right| \) would be written as " \( \ln \left(a b s\left(u^{\wedge} 3-8\right)\right) \) "

We start with the integral \[ \int \frac{5x-13}{x^{2}-5x+6} \, dx. \] **Step 1. Factor the denominator** Factor the quadratic: \[ x^{2}-5x+6=(x-2)(x-3). \] Thus the integrand becomes \[ \frac{5x-13}{(x-2)(x-3)}. \] **Step 2. Express in partial fractions** Assume the decomposition \[ \frac{5x-13}{(x-2)(x-3)} = \frac{A}{x-2} + \frac{B}{x-3}. \] Multiply both sides by \((x-2)(x-3)\): \[ 5x-13 = A(x-3) + B(x-2). \] **Step 3. Solve for coefficients** Let \(x=3\): \[ 5(3)-13 = A(3-3) + B(3-2) \quad \Rightarrow \quad 15-13 = 0 + B(1) \quad \Rightarrow \quad B=2. \] Let \(x=2\): \[ 5(2)-13 = A(2-3) + B(2-2) \quad \Rightarrow \quad 10-13 = -A \quad \Rightarrow \quad -3=-A \quad \Rightarrow \quad A=3. \] Thus, the decomposition is \[ \frac{5x-13}{(x-2)(x-3)} = \frac{3}{x-2} + \frac{2}{x-3}. \] **Step 4. Integrate each term** The integral becomes \[ \int \frac{3}{x-2} \, dx + \int \frac{2}{x-3} \, dx. \] Using the standard result \[ \int \frac{dx}{x-a} = \ln\left(abs(x-a)\right) + C, \] we have \[ \int \frac{3}{x-2}\,dx = 3\ln\left(abs(x-2)\right) \quad \text{and} \quad \int \frac{2}{x-3}\,dx = 2\ln\left(abs(x-3)\right). \] **Step 5. Write the final answer** The result of the original integral is \[ 3\ln\left(abs(x-2)\right) + 2\ln\left(abs(x-3)\right) + C. \]