∫ (x² + 4x + 2) / (6x) dx
Understand the Problem
The question is asking to solve the integral of the expression (x² + 4x + 2) / (6x) with respect to x. This involves simplifying the integrand before performing the integration.
Answer
$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
Answer for screen readers
The final result of the integration is:
$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
Steps to Solve
- Simplify the integrand
First, we will simplify the expression $\frac{x^{2} + 4x + 2}{6x}$ by breaking it into separate terms. We can rewrite it as:
$$ \frac{x^{2}}{6x} + \frac{4x}{6x} + \frac{2}{6x} $$
This simplifies to:
$$ \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} $$
- Set up the integral
Next, we set up the integral using our simplified expression:
$$ \int \left(\frac{x}{6} + \frac{2}{3} + \frac{1}{3x}\right) , dx $$
- Integrate each term
Now we can integrate each term separately:
- For $\frac{x}{6}$, the integral is:
$$ \int \frac{x}{6} , dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$
- For $\frac{2}{3}$, the integral is:
$$ \int \frac{2}{3} , dx = \frac{2}{3} x $$
- For $\frac{1}{3x}$, the integral is:
$$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln |x| $$
- Combine the results
Putting all the integrated parts together, the result is:
$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
where $C$ is the constant of integration.
The final result of the integration 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 of the function $\frac{x^{2} + 4x + 2}{6x}$ with respect to $x$. The logarithmic term arises from integrating $\frac{1}{x}$, which is a common result in calculus when dealing with rational functions.
Tips
- Ignoring the simplification step: Some may attempt to integrate the original expression directly without simplifying first, which can lead to errors.
- Error in integrating terms: Mistakes can occur when calculating the integral of each term; thus, review the power rule and the integral of logarithmic functions carefully.
