∫ (x^2 + 4x + 2) / (6x) dx
Understand the Problem
The question asks for the integral of the expression given, which involves polynomial functions in variable x. We are required to solve the integral step by step.
Answer
$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{1}{3} \ln |x| + C $$
Answer for screen readers
The final answer is
$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{1}{3} \ln |x| + C $$
Steps to Solve
- Rewrite the Integral We start with the integral
$$ \int \frac{x^2 + 4x + 2}{6x} , dx $$
This can be simplified by dividing each term in the numerator by the denominator $6x$.
- Separate the Terms We can separate the fraction into three parts:
$$ \frac{x^2}{6x} + \frac{4x}{6x} + \frac{2}{6x} $$
This simplifies to:
$$ \frac{x}{6} + \frac{4}{6} + \frac{2}{6x} $$
- Simplify the Expression Now simplifying the constants, we get:
$$ \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} $$
- Integrate Each Term Next, we will integrate each term separately:
$$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} \right) , dx $$
This results in:
- The first term becomes:
$$ \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$
- The second term becomes:
$$ \frac{2}{3} x $$
- The third term becomes:
$$ \frac{1}{3} \ln |x| $$
- Combine the Results Combining all parts gives us the final result:
$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{1}{3} \ln |x| + C $$
where $C$ is the integration constant.
The final answer is
$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{1}{3} \ln |x| + C $$
More Information
The integral combines polynomial and logarithmic functions. The process involves breaking the expression into manageable parts and using basic integration techniques.
Tips
- Forgetting to include the constant of integration $C$.
- Mixing up the integration rules, especially when dealing with logarithmic terms.
- Not simplifying the expression before integrating can lead to more complex calculations.
