∫ (x^2 + 4x^3 + 2) / (6x) dx

Steps to Solve

1. Simplify the Rational Function First, we can simplify the expression inside the integral:

$$ \frac{x^2 + 4x^3 + 2}{6x} = \frac{1}{6} \left( \frac{x^2}{x} + \frac{4x^3}{x} + \frac{2}{x} \right) $$

This gives us:

$$ \frac{1}{6} \left( x + 4x^2 + \frac{2}{x} \right) $$

2. Rewrite the Integral Now we can rewrite the integral using our simplified expression:

$$ \int \frac{x^2 + 4x^3 + 2}{6x} , dx = \frac{1}{6} \int \left( x + 4x^2 + \frac{2}{x} \right) dx $$

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

• For the first term, $x$, the integral is:

$$ \int x , dx = \frac{x^2}{2} $$

• For the second term, $4x^2$, the integral is:

$$ \int 4x^2 , dx = \frac{4x^3}{3} $$

• For the third term, $\frac{2}{x}$, the integral is:

$$ \int \frac{2}{x} , dx = 2 \ln |x| $$

Putting this all together:

$$ \int \left( x + 4x^2 + \frac{2}{x} \right) dx = \frac{x^2}{2} + \frac{4x^3}{3} + 2 \ln |x| $$

4. Combine and Scale the Result We multiply the entire integral by $\frac{1}{6}$:

$$ \frac{1}{6} \left( \frac{x^2}{2} + \frac{4x^3}{3} + 2 \ln |x| \right) = \frac{x^2}{12} + \frac{2x^3}{9} + \frac{\ln |x|}{3} + C $$

where $C$ is the constant of integration.