1. **Stating the problem:** We need to evaluate the integral $$\int \frac{dx}{x^3 + 8x + 19}$$. 2. **Understanding the integral:** The denominator is a cubic polynomial. To integrate, we typically try to factor the denominator or use partial fraction decomposition if possible. 3. **Check for roots of the cubic:** We look for rational roots using the Rational Root Theorem. Possible roots are factors of 19 (\pm1, \pm19). 4. **Evaluate the polynomial at x = -1:** $$(-1)^3 + 8(-1) + 19 = -1 - 8 + 19 = 10 \neq 0$$ 5. **Evaluate at x = 1:** $$1 + 8 + 19 = 28 \neq 0$$ 6. **Evaluate at x = -19:** $$(-19)^3 + 8(-19) + 19 = -6859 - 152 + 19 = -6992 \neq 0$$ 7. **Evaluate at x = 19:** $$19^3 + 8(19) + 19 = 6859 + 152 + 19 = 7030 \\neq 0$$ No rational roots found, so the cubic is irreducible over rationals. 8. **Use substitution or partial fractions with irreducible cubic:** Since the cubic is irreducible, the integral can be expressed in terms of partial fractions with a quadratic and linear factor or use a substitution. 9. **Try substitution:** Let $$u = x^3 + 8x + 19$$, then $$du = (3x^2 + 8) dx$$. 10. **Rewrite integral:** $$\int \frac{dx}{u}$$ but we need to express dx in terms of du and x. 11. **Express dx:** $$dx = \frac{du}{3x^2 + 8}$$. 12. **Substitute back:** Integral becomes $$\int \frac{1}{u} \cdot \frac{du}{3x^2 + 8}$$ which is complicated because of the $x$ in denominator. 13. **Conclusion:** The integral $$\int \frac{dx}{x^3 + 8x + 19}$$ does not simplify nicely with elementary functions and requires advanced methods such as partial fraction decomposition with complex roots or special functions. 14. **Final answer:** The integral cannot be expressed in elementary functions easily. It is left in integral form or evaluated numerically. **Answer:** $$\int \frac{dx}{x^3 + 8x + 19}$$