∫ (x^2 + 4x + 7) / (6x) dx

Question image

Understand the Problem

The question involves finding the integral of a rational function. Specifically, it asks for the indefinite integral of the polynomial expression divided by 6x with respect to x.

Answer

$$ \int \frac{x^2 + 4x + 7}{6x} \; dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln|x| + C $$
Answer for screen readers

$$ \int \frac{x^2 + 4x + 7}{6x} ; dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln|x| + C $$

Steps to Solve

  1. Divide the Polynomial by the Denominator

We start by simplifying the integral:

$$ \int \frac{x^2 + 4x + 7}{6x} ; dx = \int \left(\frac{x^2}{6x} + \frac{4x}{6x} + \frac{7}{6x}\right) dx $$

This simplifies to:

$$ \int \left(\frac{x}{6} + \frac{2}{3} + \frac{7}{6x}\right) dx $$

  1. Integrate Each Term Separately

Now we can integrate each term in the expression:

  • The integral of $\frac{x}{6}$ is:

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

  • The integral of $\frac{2}{3}$ is:

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

  • The integral of $\frac{7}{6x}$ is:

$$ \int \frac{7}{6x} ; dx = \frac{7}{6} \ln|x| $$

  1. Combine the Results

Putting all components together, we get:

$$ \int \frac{x^2 + 4x + 7}{6x} ; dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln|x| + C $$

where ( C ) is the constant of integration.

$$ \int \frac{x^2 + 4x + 7}{6x} ; dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln|x| + C $$

More Information

This result shows how to integrate rational functions by simplifying the expression before integrating. Each term can be treated separately, allowing for straightforward integration techniques.

Tips

  • Not Simplifying First: Some may forget to simplify the expression before integrating, complicating the process. Always simplify when possible.
  • Ignoring the Constant of Integration: It's crucial to remember to include ( C ) after performing indefinite integrals.
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