Evaluate the integral of (x^2 + 4x + 7) / (6x) dx.
Understand the Problem
The question is asking for the evaluation of the integral of the function (x^2 + 4x + 7) divided by 6x with respect to x.
Answer
\(\int \frac{x^2 + 4x + 7}{6x} \, dx = \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C\)
Answer for screen readers
The result of the integral is: [ \int \frac{x^2 + 4x + 7}{6x} , dx = \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C ]
Steps to Solve
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Divide the Function We start by breaking down the integrand: [ \frac{x^2 + 4x + 7}{6x} = \frac{x^2}{6x} + \frac{4x}{6x} + \frac{7}{6x} ] This simplifies to: [ \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} ]
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Integrate Each Term Now we can integrate each term separately: [ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} \right) dx ] Integrating each term:
- For (\frac{x}{6}), the integral is: [ \int \frac{x}{6} dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} ]
- For (\frac{2}{3}), the integral is: [ \int \frac{2}{3} dx = \frac{2}{3}x ]
- For (\frac{7}{6x}), the integral is: [ \int \frac{7}{6x} dx = \frac{7}{6} \ln |x| ]
- Combine the Results Putting it all together, we have: [ \int (x^2 + 4x + 7) / (6x) , dx = \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C ] where (C) is the constant of integration.
The result of the integral is: [ \int \frac{x^2 + 4x + 7}{6x} , dx = \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C ]
More Information
This integral involves polynomial and logarithmic function elements. The logarithmic term arises from integrating a fraction involving (x).
Tips
- Forgetting to include the constant of integration (C).
- Not simplifying each term before integrating, which can lead to mistakes.
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