∫ (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 $$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$
Answer for screen readers
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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