1. **State the problem:** We need to find the integral of the function $$\frac{x^3 + 5}{x^2 - 25}$$ with respect to $x$. 2. **Recall the formula and rules:** To integrate a rational function, it is often helpful to perform polynomial division if the degree of the numerator is greater than or equal to the degree of the denominator. 3. **Perform polynomial division:** Divide $x^3 + 5$ by $x^2 - 25$. $$x^3 + 5 \div (x^2 - 25) = x + \frac{25x + 5}{x^2 - 25}$$ 4. **Rewrite the integral:** $$\int \frac{x^3 + 5}{x^2 - 25} dx = \int \left(x + \frac{25x + 5}{x^2 - 25}\right) dx = \int x \, dx + \int \frac{25x + 5}{x^2 - 25} dx$$ 5. **Factor the denominator:** $$x^2 - 25 = (x - 5)(x + 5)$$ 6. **Use partial fraction decomposition for the second integral:** Set $$\frac{25x + 5}{(x - 5)(x + 5)} = \frac{A}{x - 5} + \frac{B}{x + 5}$$ Multiply both sides by $(x - 5)(x + 5)$: $$25x + 5 = A(x + 5) + B(x - 5)$$ 7. **Find $A$ and $B$ by substituting convenient values:** - For $x = 5$: $$25(5) + 5 = A(5 + 5) + B(0) \Rightarrow 125 + 5 = 10A \Rightarrow 130 = 10A \Rightarrow A = 13$$ - For $x = -5$: $$25(-5) + 5 = A(0) + B(-5 - 5) \Rightarrow -125 + 5 = -10B \Rightarrow -120 = -10B \Rightarrow B = 12$$ 8. **Rewrite the integral:** $$\int \frac{25x + 5}{x^2 - 25} dx = \int \frac{13}{x - 5} dx + \int \frac{12}{x + 5} dx$$ 9. **Integrate each term:** $$\int x \, dx = \frac{x^2}{2} + C_1$$ $$\int \frac{13}{x - 5} dx = 13 \ln|x - 5| + C_2$$ $$\int \frac{12}{x + 5} dx = 12 \ln|x + 5| + C_3$$ 10. **Combine all results:** $$\int \frac{x^3 + 5}{x^2 - 25} dx = \frac{x^2}{2} + 13 \ln|x - 5| + 12 \ln|x + 5| + C$$ where $C = C_1 + C_2 + C_3$ is the constant of integration. **Final answer:** $$\boxed{\int \frac{x^3 + 5}{x^2 - 25} dx = \frac{x^2}{2} + 13 \ln|x - 5| + 12 \ln|x + 5| + C}$$