∫ (x² + 4x + 2) / (6x) dx

Understand the Problem

The question is asking for the evaluation of the integral of a rational function, specifically the integral of (x² + 4x + 2) / (6x) with respect to x. This involves simplifying the expression and then applying the appropriate integration techniques.

Answer

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

Steps to Solve

Simplify the expression

To simplify the expression $\frac{x^2 + 4x + 2}{6x}$, we can separate the terms in the numerator:

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

This simplifies to:

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

Set up the integral

Now we set up the integral of the simplified expression:

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

Integrate term by term

Next, we integrate each term separately:

For $\int \frac{x}{6} , dx$:

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

For $\int \frac{2}{3} , dx$:

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

For $\int \frac{1}{3x} , dx$:

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

Combine the results

Combining all the results from the integration:

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

where $C$ is the constant of integration.

The final result of the integral is:

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

More Information

This integral represents the area under the curve defined by the rational function over a specified interval. Integrating rational functions often involves breaking them down into simpler components, which makes the integration process more manageable.

Tips

Forgetting to simplify the rational expression before integrating.
Misapplying the rules of integration, particularly for logarithmic integrals.
Not including the constant of integration $C$.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!