Evaluate the integral of (x^2 + 4x + 7) / (6x) dx.
Understand the Problem
The question is asking for the integral of a rational function, specifically the integral of (x^2 + 4x + 7) / (6x) with respect to x. The approach will involve simplifying the expression and then applying integral rules.
Answer
$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C $$
Answer for screen readers
The final result of the integral is:
$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C $$
Steps to Solve
- Simplify the integrand
We start by simplifying the expression (\frac{x^2 + 4x + 7}{6x}). We can separate the terms in the numerator:
$$ \frac{x^2}{6x} + \frac{4x}{6x} + \frac{7}{6x} $$
This simplifies to:
$$ \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} $$
- Set up the integral
Now that we have the simplified expression, we set up the integral:
$$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} \right) dx $$
- Integrate each term separately
Now we will integrate each term individually:
- 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{7}{6x}):
$$ \int \frac{7}{6x} , dx = \frac{7}{6} \ln |x| $$
- Combine the results
Putting it all together, we have:
$$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} \right) dx = \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C $$
where (C) is the constant of integration.
The final result of the integral is:
$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C $$
More Information
This result gives the antiderivative of the rational function. Integrating rational functions often involves simplifying the expression before performing the integration.
Tips
- Ignoring the simplification step can lead to complications in integration.
- Failing to separate the constant factor when integrating each term can result in errors.
