Evaluate the integral of (x² + 4x + 2)/(6x) dx.
Understand the Problem
The question is asking for the evaluation of the integral of a rational function, specifically the expression (x² + 4x + 2)/(6x) with respect to x. This involves performing polynomial long division or integration techniques.
Answer
$$ \int \frac{x^2 + 4x + 2}{6x} \, dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$
Answer for screen readers
$$ \int \frac{x^2 + 4x + 2}{6x} , dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$
Steps to Solve
- Simplify the Integral Begin by rewriting the integrand. We can separate the terms in the integrand as follows:
$$ \frac{x^2 + 4x + 2}{6x} = \frac{x^2}{6x} + \frac{4x}{6x} + \frac{2}{6x} $$
This simplifies to:
$$ \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} $$
- Set up the Integral The integral can then be expressed as the sum of three integrals:
$$ \int \left(\frac{x}{6} + \frac{2}{3} + \frac{1}{3x}\right) dx = \int \frac{x}{6} dx + \int \frac{2}{3} dx + \int \frac{1}{3x} dx $$
- Integrate each term Now we integrate each term separately:
- For the first term, apply the power rule:
$$ \int \frac{x}{6} dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$
- For the second term:
$$ \int \frac{2}{3} dx = \frac{2}{3} x $$
- For the third term, recall that the integral of $1/x$ is $\ln|x|$:
$$ \int \frac{1}{3x} dx = \frac{1}{3} \ln|x| $$
- Combine the Results Combine all the parts together to get the final result:
$$ \int \left(\frac{x^2 + 4x + 2}{6x}\right) dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$
where $C$ is the constant of integration.
$$ \int \frac{x^2 + 4x + 2}{6x} , dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$
More Information
This integral simplifies using basic rules of integration, illustrating key concepts in calculus such as breaking down rational functions and applying the power rule for integration. The constant $C$ represents the arbitrary constant of integration resulting from indefinite integration.
Tips
- Forgetting to include the constant of integration: Always remember to add $C$ to your final answer after integrating.
- Incorrectly simplifying fractions: Make sure to combine terms accurately when splitting into separate integrals.
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