∫ (x² + 4x³ + 2) / (6x) dx
Understand the Problem
The question is asking for the evaluation of a definite integral involving a polynomial function divided by another polynomial. The goal is to find the antiderivative of the given rational function with respect to x.
Answer
$$ \frac{2x^3}{9} + \frac{x^2}{12} + \frac{1}{3} \ln|x| + C $$
Answer for screen readers
$$ \int \frac{x^2 + 4x^3 + 2}{6x} , dx = \frac{2x^3}{9} + \frac{x^2}{12} + \frac{1}{3} \ln|x| + C $$
Steps to Solve
- Simplify the Integrand
First, let's simplify the expression inside the integral. We can divide each term in the numerator by the denominator:
$$ \frac{x^2 + 4x^3 + 2}{6x} = \frac{4x^3}{6x} + \frac{x^2}{6x} + \frac{2}{6x} $$
This simplifies to:
$$ \frac{4x^2}{6} + \frac{x}{6} + \frac{1}{3x} $$
- Rewrite the Integral
Now that we have simplified the integrand, we can rewrite the integral:
$$ \int \left( \frac{4}{6}x^2 + \frac{1}{6}x + \frac{1}{3x} \right) , dx $$
This can be further simplified to:
$$ \int \left( \frac{2}{3}x^2 + \frac{1}{6}x + \frac{1}{3x} \right) , dx $$
- Integrate Each Term Separately
Now, we will integrate each term separately:
- For $\frac{2}{3}x^2$, the integral is:
$$ \int \frac{2}{3}x^2 , dx = \frac{2}{3} \cdot \frac{x^3}{3} = \frac{2x^3}{9} $$
- For $\frac{1}{6}x$, the integral is:
$$ \int \frac{1}{6}x , dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$
- For $\frac{1}{3x}$, the integral is:
$$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln|x| $$
- Combine the Results
Putting it all together, we get:
$$ \int \frac{x^2 + 4x^3 + 2}{6x} , dx = \frac{2x^3}{9} + \frac{x^2}{12} + \frac{1}{3} \ln|x| + C $$
where (C) is the constant of integration.
$$ \int \frac{x^2 + 4x^3 + 2}{6x} , dx = \frac{2x^3}{9} + \frac{x^2}{12} + \frac{1}{3} \ln|x| + C $$
More Information
This integral combines polynomial and logarithmic functions. Definite integrals involving rational functions often require decomposition or simplification for easier integration. Understanding antiderivatives is fundamental in calculus.
Tips
- Forgetting to include the constant of integration (C) when finding indefinite integrals.
- Not simplifying the rational function before integrating, which can lead to more complex calculations.