∫ (x^2 + 4x + 2) / (6x) dx
Understand the Problem
The question is asking for the integral of the function (x^2 + 4x + 2) / (6x) with respect to x. We will solve this by simplifying the expression and then performing the integration.
Answer
The integral is $$ \frac{1}{12} x^2 + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
Answer for screen readers
The integral is
$$ \frac{1}{12} x^2 + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
Steps to Solve
- Simplify the integrand
We start by simplifying the expression ( \frac{x^2 + 4x + 2}{6x} ). We can split the fraction into separate terms:
$$ \frac{x^2}{6x} + \frac{4x}{6x} + \frac{2}{6x} $$
This simplifies to:
$$ \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} $$
- Rewrite the integral
Now we can rewrite the integral with the simplified expression:
$$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} \right) dx $$
- Integrate term by term
Next, we will integrate each term individually:
- The integral of ( \frac{x}{6} ) is:
$$ \int \frac{x}{6} , dx = \frac{1}{12} x^2 $$
- The integral of ( \frac{2}{3} ) is:
$$ \int \frac{2}{3} , dx = \frac{2}{3} x $$
- The integral of ( \frac{1}{3x} ) is:
$$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln |x| $$
- Combine the results
Combining these results, we get:
$$ \int (x^2 + 4x + 2)/(6x) , dx = \frac{1}{12} x^2 + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
where ( C ) is the constant of integration.
The integral is
$$ \frac{1}{12} x^2 + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
More Information
This integral represents the area under the curve of the function ( \frac{x^2 + 4x + 2}{6x} ) and is useful in various applications, such as finding the total distance traveled by an object over time.
Tips
- Forgetting to split the fraction before integrating.
- Misapplying the integral rules, especially for constants and logarithmic functions.
