∫ (x² + 4x³ + 2) / (6x) dx

Question image

Understand the Problem

The question is asking to evaluate the integral of the expression (x² + 4x³ + 2) / (6x) with respect to x. This involves simplifying the fraction and then performing the integration.

Answer

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

The integral evaluates to:

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

Steps to Solve

  1. Simplify the integrand

    First, simplify the fraction:

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

    This can be simplified further:

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

  2. Set up the integral

    Now, rewrite the integral with the simplified expression:

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

  3. Integrate each term

    Now, integrate each term:

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

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

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

  4. Combine the results

    Combine the results of the integrals:

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

    where (C) is the constant of integration.

The integral evaluates to:

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

More Information

This result shows the integration of a polynomial divided by a linear term and involves power and logarithmic integration techniques. The constant (C) represents any constant since the original function can have various antiderivatives.

Tips

  • Failing to simplify the fraction properly before integration. Always simplify first to make integration easier.
  • Not remembering the integral of (\frac{1}{x}), which is (\ln|x|). Ensure you apply the correct rules for logarithmic integration.

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