∫ (x² + 4x³ + 2) / (6x) dx

Steps to Solve

1. Simplify the Expression
   We start by rewriting the integrand.
   Divide each term of the numerator by the denominator: $$ \frac{x^2}{6x} + \frac{4x^3}{6x} + \frac{2}{6x} = \frac{x}{6} + \frac{2}{3}x^2 + \frac{1}{3x} $$

2. Set Up the Integral
   Now we can set up the integral with the simplified expression:
   $$ \int \left( \frac{x}{6} + \frac{2}{3}x^2 + \frac{1}{3x} \right) dx $$

3. Integrate Each Term
   We will integrate each term separately:

• The integral of $\frac{x}{6}$ is $\frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12}$.
• The integral of $\frac{2}{3}x^2$ is $\frac{2}{3} \cdot \frac{x^3}{3} = \frac{2}{9}x^3$.
• The integral of $\frac{1}{3x}$ is $\frac{1}{3} \ln |x|$.

Putting it all together, we have: $$ \int \left( \frac{x}{6} + \frac{2}{3}x^2 + \frac{1}{3x} \right) dx = \frac{x^2}{12} + \frac{2}{9}x^3 + \frac{1}{3} \ln |x| + C $$

4. Final Result
   The final result of the integration is: $$ \frac{x^2}{12} + \frac{2}{9}x^3 + \frac{1}{3} \ln |x| + C $$