Use a change of variables to evaluate the following definite integral. \( \int_{1}^{2} \frac{5}{9 x^{2}+12 x+4} d x \) Determine a change of variables from \( x \) to \( u \). Choose the correct answer below. A. \( u=9 x^{2}+12 x+4 \) C. \( u=3 x+2 \) Dr. \( u=3 x \) Write the integral in terms of \( u \). 2 \( \int_{1}^{2} \frac{5}{9 x^{2}+12 x+4} d x=\int_{5}(\square) d u \) Evaluate the integral. 2 \( \int_{1}^{2} \frac{5}{9 x^{2}+12 x+4} d x=\square \) (Type an exact answer.) B

Let \( u = 3x + 2 \). Notice that \[ (3x+2)^2 = 9x^2 + 12x + 4, \] so this substitution transforms the denominator perfectly. Differentiate \( u \) with respect to \( x \): \[ \frac{du}{dx} = 3 \quad \Rightarrow \quad dx = \frac{du}{3}. \] Next, change the limits of integration. When \( x = 1 \): \[ u = 3(1) + 2 = 5, \] and when \( x = 2 \): \[ u = 3(2) + 2 = 8. \] Now, substitute into the integral: \[ \int_{1}^{2} \frac{5}{9x^2+12x+4}\, dx = \int_{5}^{8} \frac{5}{(3x+2)^2} \cdot \frac{du}{3}. \] Since \( 9x^2+12x+4 = (3x+2)^2 = u^2 \), the integral becomes: \[ \frac{5}{3} \int_{5}^{8} \frac{1}{u^2}\, du. \] Now, compute the integral: \[ \int \frac{1}{u^2}\, du = \int u^{-2}\, du = -u^{-1} + C = -\frac{1}{u} + C. \] Evaluate from \( u = 5 \) to \( u = 8 \): \[ \frac{5}{3} \left[-\frac{1}{u}\right]_{5}^{8} = \frac{5}{3} \left(-\frac{1}{8} + \frac{1}{5}\right). \] Simplify the expression inside the parentheses: \[ -\frac{1}{8} + \frac{1}{5} = \frac{-5 + 8}{40} = \frac{3}{40}. \] Thus, the integral is: \[ \frac{5}{3} \cdot \frac{3}{40} = \frac{5}{40} = \frac{1}{8}. \] The final result is: \[ \boxed{\frac{1}{8}}. \]