∫ (x^2 + 4x + 7) / (6x) dx
Understand the Problem
The question is asking to solve the integral of the given rational function, which involves applying integration techniques, likely partial fraction decomposition or substitution.
Answer
$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln |x| + C $$
Answer for screen readers
The final answer is
$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln |x| + C $$
Steps to Solve
- Rewrite the Integral We start with the integral
$$ \int \frac{x^2 + 4x + 7}{6x} , dx $$
This can be simplified by dividing each term in the numerator by (6x):
$$ \int \left( \frac{x^2}{6x} + \frac{4x}{6x} + \frac{7}{6x} \right) dx = \int \left( \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} \right) dx $$
- Split the Integral Now, we can express the integral as a sum of integrals:
$$ \int \left( \frac{x}{6} \right) dx + \int \left( \frac{2}{3} \right) dx + \int \left( \frac{7}{6x} \right) dx $$
- Compute Each Integral Calculate each integral separately:
- For the first term:
$$ \int \frac{x}{6} , dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$
- For the second term:
$$ \int \frac{2}{3} , dx = \frac{2}{3} x $$
- For the third term:
$$ \int \frac{7}{6x} , dx = \frac{7}{6} \ln |x| $$
- Combine the Results Now, combine all the results from the three integrals:
$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln |x| + C $$
where (C) is the constant of integration.
The final answer is
$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln |x| + C $$
More Information
This integral leverages basic integration techniques including power rules and the natural logarithm of (x). Such integrals are common in calculus, especially with rational functions.
Tips
- Forgetting to include the constant of integration (C) at the end.
- Miscalculating integral components such as ( \int \frac{1}{x} , dx ); it should yield ( \ln |x| ).
- Not simplifying the rational function before integrating, which can complicate the process unnecessarily.
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