∫ (x² + 4x³ + 2) / (6x) dx
Understand the Problem
The question is asking for the solution to the integral of a rational function, specifically the integral of (x² + 4x³ + 2) / (6x) with respect to x. This will involve simplifying the expression and applying integration techniques.
Answer
$$ \int \frac{x^{2} + 4x^{3} + 2}{6x} \, dx = \frac{x^{2}}{12} + \frac{2x^{3}}{9} + \frac{1}{3} \ln|x| + C $$
Answer for screen readers
The final answer is:
$$ \int \frac{x^{2} + 4x^{3} + 2}{6x} , dx = \frac{x^{2}}{12} + \frac{2x^{3}}{9} + \frac{1}{3} \ln|x| + C $$
Steps to Solve
- Simplify the Expression
To simplify the integral, divide each term in the numerator by the denominator. The expression becomes:
$$ \frac{x^{2}}{6x} + \frac{4x^{3}}{6x} + \frac{2}{6x} = \frac{x}{6} + \frac{2}{3}x^{2} + \frac{1}{3x} $$
- Rewrite the Integral
Now, rewrite the integral using the simplified expression:
$$ \int \left( \frac{x}{6} + \frac{2}{3}x^{2} + \frac{1}{3x} \right) dx $$
- Integrate Each Term Separately
Now, integrate each term one by one:
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For $\frac{x}{6}$: $$ \int \frac{x}{6} , dx = \frac{1}{6} \cdot \frac{x^{2}}{2} = \frac{x^{2}}{12} $$
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For $\frac{2}{3}x^{2}$: $$ \int \frac{2}{3} x^{2} , dx = \frac{2}{3} \cdot \frac{x^{3}}{3} = \frac{2x^{3}}{9} $$
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For $\frac{1}{3x}$: $$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln|x| $$
- Combine the Results
Combine the results of the integrals from the previous step:
$$ \int \frac{x^{2} + 4x^{3} + 2}{6x} , dx = \frac{x^{2}}{12} + \frac{2x^{3}}{9} + \frac{1}{3} \ln|x| + C $$
where $C$ is the constant of integration.
The final answer is:
$$ \int \frac{x^{2} + 4x^{3} + 2}{6x} , dx = \frac{x^{2}}{12} + \frac{2x^{3}}{9} + \frac{1}{3} \ln|x| + C $$
More Information
This integral involves simplifying a rational function and applying the rules of integration term by term. It's essential to remember to add the constant of integration, $C$, at the end of the calculation, as it accounts for any constant value that could have been present in the original function.
Tips
- Forgetting to simplify the rational expression before integrating.
- Not including the constant of integration $C$ in the final answer.
- Misapplying integration rules for logarithmic functions.
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