∫ (x² + 4x³ + 7) / (6x) dx
Understand the Problem
The question is asking for the integration of the expression (x^2 + 4x^3 + 7) / (6x) with respect to x. This involves simplifying the expression and then performing the integration step by step.
Answer
The integral of the expression is $$ \frac{1}{12} x^2 + \frac{2}{9} x^3 + \frac{7}{6} \ln|x| + C $$
Answer for screen readers
The final answer to the integral is
$$ \frac{1}{12} x^2 + \frac{2}{9} x^3 + \frac{7}{6} \ln|x| + C $$
Steps to Solve
- Simplify the Expression
We start by simplifying the fraction:
$$ \frac{x^2 + 4x^3 + 7}{6x} = \frac{x^2}{6x} + \frac{4x^3}{6x} + \frac{7}{6x} $$
This simplifies to:
$$ \frac{1}{6}x + \frac{2}{3}x^2 + \frac{7}{6x} $$
- Set Up the Integral
Now, we set up the integral of the simplified expression:
$$ \int \left(\frac{1}{6}x + \frac{2}{3}x^2 + \frac{7}{6x}\right) dx $$
- Integrate Each Term
Now we integrate each term separately:
- For the first term:
$$ \int \frac{1}{6} x , dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{1}{12} x^2 $$
- For the second term:
$$ \int \frac{2}{3} x^2 , dx = \frac{2}{3} \cdot \frac{x^3}{3} = \frac{2}{9} x^3 $$
- For the third term:
$$ \int \frac{7}{6} \cdot \frac{1}{x} , dx = \frac{7}{6} \ln|x| $$
- Combine the Results
Combining the integrals gives us:
$$ \frac{1}{12} x^2 + \frac{2}{9} x^3 + \frac{7}{6} \ln|x| + C $$
where (C) is the constant of integration.
The final answer to the integral is
$$ \frac{1}{12} x^2 + \frac{2}{9} x^3 + \frac{7}{6} \ln|x| + C $$
More Information
The integral combines polynomial functions and a logarithmic function, and the presence of the variable in the denominator introduces the natural logarithm when integrated.
Tips
- Forgetting to include the constant of integration (C).
- Simplifying the expression incorrectly before integration.
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