∫ (x^2 + 4x + 2) / (6x) dx
Understand the Problem
The question is asking to find the integral of the given rational function, which involves polynomial terms in the numerator and denominator.
Answer
$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{1}{3} \ln|x| + C $$
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
-
Rewrite the Integral
Rewrite the integral for clarity.
$$ \int \frac{x^2 + 4x + 2}{6x} , dx $$ -
Divide Each Term
Separate the terms in the numerator from the denominator.
$$ \int \left( \frac{x^2}{6x} + \frac{4x}{6x} + \frac{2}{6x} \right) , dx $$
This simplifies to:
$$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} \right) , dx $$ -
Integrate Each Term
Now integrate each term separately:
$$ \int \frac{x}{6} , dx + \int \frac{2}{3} , dx + \int \frac{1}{3x} , dx $$ -
Calculate the Integrals
- For $\int \frac{x}{6} , dx$:
$$ \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$ - For $\int \frac{2}{3} , dx$:
$$ \frac{2}{3}x $$ - For $\int \frac{1}{3x} , dx$:
$$ \frac{1}{3} \ln|x| $$
-
Combine the Results
Combine the integrated results:
$$ \frac{x^2}{12} + \frac{2}{3}x + \frac{1}{3} \ln|x| + C $$
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 result of the integration includes a polynomial term and a logarithmic term, representing a common scenario when integrating rational functions. The constant ( C ) represents the integration constant, which is added because the indefinite integral can vary by a constant value.
Tips
- Neglecting to simplify the fraction correctly before integrating.
- Failing to include the constant of integration ( C ).
