∫ (x² + 4x³ + 2) / (6x) dx

Steps to Solve

1. Simplify the Integrand

First, let's simplify the expression inside the integral. We can divide each term in the numerator by the denominator:

$$ \frac{x^2 + 4x^3 + 2}{6x} = \frac{4x^3}{6x} + \frac{x^2}{6x} + \frac{2}{6x} $$

This simplifies to:

$$ \frac{4x^2}{6} + \frac{x}{6} + \frac{1}{3x} $$

2. Rewrite the Integral

Now that we have simplified the integrand, we can rewrite the integral:

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

This can be further simplified to:

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

3. Integrate Each Term Separately

Now, we will integrate each term separately:

• For $\frac{2}{3}x^2$, the integral is:

$$ \int \frac{2}{3}x^2 , dx = \frac{2}{3} \cdot \frac{x^3}{3} = \frac{2x^3}{9} $$

• For $\frac{1}{6}x$, the integral is:

$$ \int \frac{1}{6}x , dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$

• For $\frac{1}{3x}$, the integral is:

$$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln|x| $$

4. Combine the Results

Putting it all together, we get:

$$ \int \frac{x^2 + 4x^3 + 2}{6x} , dx = \frac{2x^3}{9} + \frac{x^2}{12} + \frac{1}{3} \ln|x| + C $$

where (C) is the constant of integration.