∫ (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: $$ \frac{x^2}{12} + \frac{2}{3}x + \frac{1}{3} \ln |x| + C $$
Answer for screen readers
The final result of the integral is:
$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{1}{3} \ln |x| + C $$
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 $C$.
