Letermine the following indefinite integral. Check your work by differentiation. \[ \int \frac{5}{9 r^{2}+9} d r \]

To evaluate the indefinite integral \[ \int \frac{5}{9 r^{2} + 9} \, dr, \] let's simplify the integrand first. **Step 1: Factor the denominator** \[ 9 r^{2} + 9 = 9(r^{2} + 1). \] So the integral becomes: \[ \int \frac{5}{9(r^{2} + 1)} \, dr = \frac{5}{9} \int \frac{1}{r^{2} + 1} \, dr. \] **Step 2: Integrate using the standard integral** We know that \[ \int \frac{1}{r^{2} + 1} \, dr = \tan^{-1}(r) + C, \] where \( C \) is the constant of integration. **Step 3: Combine the constants** \[ \frac{5}{9} \int \frac{1}{r^{2} + 1} \, dr = \frac{5}{9} \tan^{-1}(r) + C. \] **Final Answer:** \[ \frac{5}{9} \tan^{-1}(r) + C \] **Verification by Differentiation:** Differentiate the result to ensure it matches the original integrand: \[ \frac{d}{dr} \left( \frac{5}{9} \tan^{-1}(r) + C \right) = \frac{5}{9} \cdot \frac{1}{1 + r^{2}} = \frac{5}{9(r^{2} + 1)}. \] This matches the original integrand, confirming that the integration is correct. **Answer:** After simplifying, an antiderivative is (5⁄9) arctan r plus a constant. Thus, ∫5/(9r²+9) dr = (5/9) tan⁻¹(r) + C