∫ (x² + 4x + 2)/(6x) dx

Understand the Problem

The question is asking for the integral of the expression (x² + 4x + 2)/(6x) with respect to x. This involves performing the integration process, which may include simplifying the expression before integrating.

Answer

$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$

Steps to Solve

Simplify the expression

We start by dividing each term in the numerator by the denominator. The integral can be rewritten as:

$$ \int \left( \frac{x^2}{6x} + \frac{4x}{6x} + \frac{2}{6x} \right) dx $$

This simplifies to:

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

Integrate each term separately

Now we can integrate each term in the expression separately:

For $\frac{x}{6}$:

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

For $\frac{2}{3}$:

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

For $\frac{1}{3x}$:

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

Combine the results

Now we combine the results of the integrals:

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

Where (C) is the constant of integration.

The final answer for the integral is:

$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$

More Information

This integral involves basic polynomial and logarithmic functions. Such integration techniques are commonly used in calculus, particularly in finding areas under curves and solving differential equations.

Tips

Forgetting to apply the constant of integration (C) at the end is a common mistake.
Not properly simplifying the rational expression before integrating can lead to incorrect results.

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