∫ (x² + 4x + 2) / (6x) dx

Question image

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

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

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

Steps to Solve

  1. 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:

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

  1. Set Up the Integral

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

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

  1. 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:

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

  1. Write the Final Result

Thus, the final result of the integral is:

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

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

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