∫ (x² + 4x³ + 2) / (6x) dx

Question image

Understand the Problem

The question is asking to solve the integral of a rational function. We have to integrate (x² + 4x³ + 2) / (6x) with respect to x.

Answer

$$ \int \frac{x^2 + 4x^3 + 2}{6x} \, dx = \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$
Answer for screen readers

The integral is given by:

$$ \int \frac{x^2 + 4x^3 + 2}{6x} , dx = \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

Steps to Solve

  1. Simplify the integrand

First, we divide each term in the numerator by the denominator $6x$:

$$ \frac{x^2}{6x} + \frac{4x^3}{6x} + \frac{2}{6x} $$

This simplifies to:

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

  1. Separate the integral

Next, express the integral as the sum of three separate integrals:

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

  1. Compute each integral

Now, we compute each integral individually:

  • For the first term:

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

  • For the second term:

$$ \int \frac{2}{3}x^2 dx = \frac{2}{3} \cdot \frac{x^3}{3} = \frac{2x^3}{9} $$

  • For the third term:

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

  1. Combine the results

The final integral result combines all computed integrals and includes a constant of integration $C$:

$$ \int \frac{x^2 + 4x^3 + 2}{6x} dx = \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

The integral is given by:

$$ \int \frac{x^2 + 4x^3 + 2}{6x} , dx = \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

More Information

This result demonstrates the application of basic integral calculus involving rational functions. When integrating, each term can be treated separately, simplifying calculations.

Tips

  • Forgetting to separate the integral into simpler parts.
  • Misapplying the power rule for integration, particularly with constants.
  • Ignoring the absolute value in the logarithm term.
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