∫(x² + 4x³ + 2) / (6x) dx
Understand the Problem
The question involves finding the integral of the expression (x² + 4x³ + 2) divided by 6x with respect to x. This involves applying integration techniques for rational functions.
Answer
$$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$
Answer for screen readers
The final answer is
$$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$
Steps to Solve
- Simplify the integrand
We start with the expression
$$ \frac{x^2 + 4x^3 + 2}{6x} $$
To simplify, divide each term in the numerator by the denominator:
$$ \frac{x^2}{6x} + \frac{4x^3}{6x} + \frac{2}{6x} = \frac{x}{6} + \frac{2}{3}x^2 + \frac{1}{3x} $$
So the integral becomes:
$$ \int \left( \frac{x}{6} + \frac{2}{3}x^2 + \frac{1}{3x} \right) dx $$
- Integrate each term
Now we integrate each term 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,
$$ \int \frac{2}{3}x^2 , dx = \frac{2}{3} \cdot \frac{x^3}{3} = \frac{2x^3}{9} $$
- For the third term,
$$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln |x| $$
Putting it all together, we have:
$$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$
- Write the final result
Combine the results into the final answer:
$$ \int \frac{x^2 + 4x^3 + 2}{6x} , dx = \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$
The final answer is
$$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$
More Information
This result shows how to handle rational functions by simplifying them before integration. Each term integrated separately makes it easier to apply basic integration techniques.
Tips
- Forgetting to simplify the rational expression before integrating.
- Mixing up constants when integrating, especially with logarithmic terms.
- Neglecting the absolute value in the logarithmic term for $x$.
