∫ (x² + 4x + 2) / (6x) dx

Understand the Problem

The question is asking for the integral of the expression (x² + 4x + 2) / (6x) with respect to x. To solve it, we need to simplify the expression, potentially break it into simpler parts, and then integrate term by term.

Answer

The integral is $$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$

Steps to Solve

Simplify the Expression

We start by dividing each term in the numerator by the denominator:

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

This simplifies to:

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

Set Up the Integral

Now we can express the integral as:

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

Integrate Each Term

Integrate each term separately:

For the first term, we have:

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

For the second term:

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

For the third term:

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

Combine Results

Now we combine the integrated terms and add the constant of integration $C$:

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

The integral is

$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$

More Information

This integral is a straightforward example of using basic integration rules. It's broken down into simple terms that can be easily integrated, showcasing the power of simplification in calculus.

Tips

Forgetting to include the constant of integration $C$ after solving the integral.
Misapplying the rules of integration for logarithmic functions, especially $\int \frac{1}{x} , dx$.

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