Evaluate the integral of (x^2 + 4x + 7) / (6x) dx.

Question image

Understand the Problem

The question is asking for the evaluation of the integral of the function (x^2 + 4x + 7) divided by 6x with respect to x.

Answer

\(\int \frac{x^2 + 4x + 7}{6x} \, dx = \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C\)
Answer for screen readers

The result of the integral is: [ \int \frac{x^2 + 4x + 7}{6x} , dx = \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C ]

Steps to Solve

  1. Divide the Function We start by breaking down the integrand: [ \frac{x^2 + 4x + 7}{6x} = \frac{x^2}{6x} + \frac{4x}{6x} + \frac{7}{6x} ] This simplifies to: [ \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} ]

  2. Integrate Each Term Now we can integrate each term separately: [ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} \right) dx ] Integrating each term:

  • For (\frac{x}{6}), the integral is: [ \int \frac{x}{6} dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} ]
  • For (\frac{2}{3}), the integral is: [ \int \frac{2}{3} dx = \frac{2}{3}x ]
  • For (\frac{7}{6x}), the integral is: [ \int \frac{7}{6x} dx = \frac{7}{6} \ln |x| ]
  1. Combine the Results Putting it all together, we have: [ \int (x^2 + 4x + 7) / (6x) , dx = \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C ] where (C) is the constant of integration.

The result of the integral is: [ \int \frac{x^2 + 4x + 7}{6x} , dx = \frac{x^2}{12} + \frac{2}{3}x + \frac{7}{6} \ln |x| + C ]

More Information

This integral involves polynomial and logarithmic function elements. The logarithmic term arises from integrating a fraction involving (x).

Tips

  • Forgetting to include the constant of integration (C).
  • Not simplifying each term before integrating, which can lead to mistakes.

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