∫ (x² + 4x + 2) / (6x) dx

Understand the Problem

The question involves calculating the integral of a rational function expressed as the fraction of a polynomial divided by another polynomial, specifically the integral of (x² + 4x + 2) / (6x) with respect to x.

Answer

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

Steps to Solve

Simplify the Integrand

First, we will simplify the expression $\frac{x^2 + 4x + 2}{6x}$. We can separate the terms in the numerator:

$$ \frac{x^2}{6x} + \frac{4x}{6x} + \frac{2}{6x} $$

This simplifies to:

$$ \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} $$

Set Up the Integral

Now, we can set up the integral based on our simplified expression:

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

Integrate Each Term

Next, we will integrate each term separately:

The integral of $\frac{x}{6}$ is $\frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12}$.
The integral of $\frac{2}{3}$ is $\frac{2}{3}x$.
The integral of $\frac{1}{3x}$ is $\frac{1}{3} \ln|x|$.

Therefore, the combined integral is:

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

Write the Final Result

Thus, the final result of the integral is:

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

More Information

This integral represents the area under the curve of the function $\frac{x^2 + 4x + 2}{6x}$ over a specified interval. Integrating rational functions often involves simplifying the integrand to make it easier to handle, which is a common technique in calculus.

Tips

Forgetting to separate the terms in the numerator when simplifying the integrand.
Miscalculating the integrals of individual terms, especially logarithmic functions.
Not including the constant of integration (C).

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