∫ (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. This involves performing the integration process, which may include simplifying the expression before integrating.
Answer
$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
Answer for screen readers
The final answer for 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. The integral can be rewritten as:
$$ \int \left( \frac{x^2}{6x} + \frac{4x}{6x} + \frac{2}{6x} \right) dx $$
This simplifies to:
$$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} \right) dx $$
- Integrate each term separately
Now we can integrate each term in the expression separately:
- For $\frac{x}{6}$:
$$ \int \frac{x}{6} , dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$
- For $\frac{2}{3}$:
$$ \int \frac{2}{3} , dx = \frac{2}{3} x $$
- For $\frac{1}{3x}$:
$$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln |x| $$
- Combine the results
Now we combine the results of the integrals:
$$ \int \left( \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| \right) dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
Where (C) is the constant of integration.
The final answer for the integral is:
$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
More Information
This integral involves basic polynomial and logarithmic functions. Such integration techniques are commonly used in calculus, particularly in finding areas under curves and solving differential equations.
Tips
- Forgetting to apply the constant of integration (C) at the end is a common mistake.
- Not properly simplifying the rational expression before integrating can lead to incorrect results.
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