∫ (x² + 4x + 2) / (6x) dx
Understand the Problem
The question involves integrating a rational function, specifically the fraction formed by the polynomial (x² + 4x + 2) divided by 6x. The goal is to find the integral of this expression with respect to x.
Answer
The integral is $$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
Answer for screen readers
The integral is given by: $$ \int \frac{x^2 + 4x + 2}{6x} , dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
Steps to Solve
-
Simplify the Fraction Start by dividing each term in the numerator by the denominator: $$ \frac{x^2 + 4x + 2}{6x} = \frac{x^2}{6x} + \frac{4x}{6x} + \frac{2}{6x} $$ This simplifies to: $$ \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} $$
-
Set Up the Integral Now set up the integral with the simplified expression: $$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} \right) , dx $$
-
Integrate Each Term Integrate each term separately:
-
For $\frac{x}{6}$: $$ \int \frac{x}{6} , dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12} $$
-
For $\frac{2}{3}$: $$ \int \frac{2}{3} , dx = \frac{2}{3} x $$
-
For $\frac{1}{3x}$: $$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln |x| $$
-
-
Combine the Results Combine all the integrated terms: $$ \int \frac{x^2 + 4x + 2}{6x} , dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$ where $C$ is the constant of integration.
The integral is given by: $$ \int \frac{x^2 + 4x + 2}{6x} , dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C $$
More Information
The result represents the antiderivative of the rational function. The constant $C$ accounts for all possible constants that could be added to the function without changing its derivative.
Tips
- Forgetting to simplify the fraction before integrating.
- Omitting the constant of integration $C$ in the final result.
AI-generated content may contain errors. Please verify critical information
