Integrate (x² + 4x³ + 2) / (6x) dx

Steps to Solve

1. Rewrite the Rational Function
   We can simplify the integral by dividing each term in the numerator by the denominator. The function becomes:
   $$ \int \frac{x^2}{6x} + \frac{4x^3}{6x} + \frac{2}{6x} , dx $$
   Which simplifies to:
   $$ \int \frac{x}{6} + \frac{2}{3}x^2 + \frac{1}{3x} , dx $$

2. Separate the Integral
   Next, we can separate the integral into individual terms:
   $$ \int \left( \frac{x}{6} + \frac{2}{3} x^2 + \frac{1}{3x} \right) , dx = \frac{1}{6} \int x , dx + \frac{2}{3} \int x^2 , dx + \frac{1}{3} \int \frac{1}{x} , dx $$

3. Integrate Each Term
   Now we will integrate each term individually:

• For the first term:
  $$ \int x , dx = \frac{x^2}{2} $$
• For the second term:
  $$ \int x^2 , dx = \frac{x^3}{3} $$
• For the third term:
  $$ \int \frac{1}{x} , dx = \ln |x| $$

4. Combine the Results
   Now, substituting back into our equation: $$ \frac{1}{6} \cdot \frac{x^2}{2} + \frac{2}{3} \cdot \frac{x^3}{3} + \frac{1}{3} \ln |x| + C $$ This simplifies to:
   $$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

5. Final Result
   The final result of the integral is:
   $$ \int \frac{x^2 + 4x^3 + 2}{6x} , dx = \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$