∫ (x^2 + 4x^3 + 2) / (6x) dx

Question image

Understand the Problem

The question is asking to evaluate the integral of the expression given above, which involves polynomial terms in the numerator divided by a linear term in the denominator.

Answer

The evaluated integral is $$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$
Answer for screen readers

The evaluated integral is

$$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

Steps to Solve

  1. Rewrite the Integral

First, simplify the expression by dividing each term in the numerator by the denominator $6x$:

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

This simplifies to:

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

  1. Separate the Integral

Now separate the integral into individual terms:

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

  1. Calculate Each Integral

Now calculate each integral one by one:

  • 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:

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

  • For the third term:

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

  1. Combine the Results

Now combine the results of the integrals:

$$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

where $C$ is the constant of integration.

The evaluated integral is

$$ \frac{x^2}{12} + \frac{2x^3}{9} + \frac{1}{3} \ln |x| + C $$

More Information

This integral involves simplifying a rational function before integrating, which is a common technique in calculus. The results combine polynomial terms and a logarithmic term, reflecting the different forms of integration.

Tips

  • Misdividing the terms in the numerator when simplifying.
  • Forgetting to include the constant of integration $C$.
  • Confusing the integration rules for polynomials and logarithmic functions.
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