∫ (x^2 + 4x^3 + 2) / (6x) dx

Understand the Problem

The question is asking to solve the integral of a polynomial function divided by a linear function. The integral involved here requires step-by-step calculations.

Answer

The solution to the integral is: $$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

Steps to Solve

Simplify the expression

To simplify the integral, we divide each term in the numerator by the denominator:

$$ \frac{x^2}{6x} + \frac{4x^3}{6x} + \frac{2}{6x} = \frac{x}{6} + \frac{2}{3}x^2 + \frac{1}{3x} $$

Set up the integral

Now rewrite the integral with the simplified expression:

$$ \int \left( \frac{x}{6} + \frac{2}{3}x^2 + \frac{1}{3x} \right) dx $$

Integrate each term separately

Next, integrate each term one by one:

For ( \frac{x}{6} ):

$$ \int \frac{x}{6} , dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$

For ( \frac{2}{3}x^2 ):

$$ \int \frac{2}{3}x^2 , dx = \frac{2}{3} \cdot \frac{x^3}{3} = \frac{2x^3}{9} $$

For ( \frac{1}{3x} ):

$$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln |x| $$

Combine the results

Now, combine all the integrated terms:

$$ \int \left( \frac{x}{6} + \frac{2}{3}x^2 + \frac{1}{3x} \right) dx = \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

where ( C ) is the constant of integration.

The integral evaluates to:

$$ \int \frac{x^2 + 4x^3 + 2}{6x} , dx = \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

More Information

This integral combines polynomial and logarithmic terms. It illustrates the use of basic integration techniques and simplifies the more complex rational function step-by-step. The use of the natural logarithm arises from integrating the function ( \frac{1}{x} ).

Tips

Forgetting to simplify the fraction before integrating.
Mixing up the constant during integration or omitting the constant ( C ).
Failing to recognize the form of ( \frac{1}{x} ) that leads to a logarithm.

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