∫ (x^2 + 4x + 2) / (6x) dx
Understand the Problem
The question is asking to solve the integral of the expression given, which is a fraction involving polynomial terms in the numerator and denominator. The goal is to find the antiderivative of the given function with respect to x.
Answer
$$ \int \frac{x^2 + 4x + 2}{6x} dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$
Answer for screen readers
The final answer is:
$$ \int \frac{x^2 + 4x + 2}{6x} dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$
Steps to Solve
- Simplify the Integral
To simplify the expression, divide each term in the numerator by the term in the denominator:
$$ \frac{x^2 + 4x + 2}{6x} = \frac{x^2}{6x} + \frac{4x}{6x} + \frac{2}{6x} $$
This simplifies to:
$$ \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} $$
So, we rewrite the integral as:
$$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} \right) dx $$
- Integrate Each Term Separately
Now, we can integrate each term in the expression separately:
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For the first term, $\int \frac{x}{6} dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12}$
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For the second term, $\int \frac{2}{3} dx = \frac{2}{3} x$
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For the third term, $\int \frac{1}{3x} dx = \frac{1}{3} \ln|x|$
Combining these results, we have:
$$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} \right) dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| $$
- Add the Constant of Integration
Don’t forget to add the constant of integration (C):
$$ \int \frac{x^2 + 4x + 2}{6x} dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$
The final answer is:
$$ \int \frac{x^2 + 4x + 2}{6x} dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln|x| + C $$
More Information
The integral we solved involves breaking down the fraction into simpler terms for easier integration. Each term was integrated separately, which is a common technique in integral calculus. The constant of integration represents an arbitrary constant added to indefinite integrals.
Tips
- Neglecting to simplify first: Always simplify the fraction before attempting to integrate.
- Forgetting the constant of integration: Always add the constant (C) at the end of the integration.
- Misapplying logarithmic integration: Remember that for $ \int \frac{1}{x} dx = \ln |x| + C$.
