∫ (x² + 4x + 2)/(6x) dx

Question image

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

$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
Answer for screen readers

The final answer for the integral is:

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

Steps to Solve

  1. 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:

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

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

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

  1. 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.
Thank you for voting!
Use Quizgecko on...
Browser
Browser