∫ (x^2 + 4x + 2) / (6x) dx
Understand the Problem
The question involves evaluating the integral of a rational function, specifically the integral of (x^2 + 4x + 2) / (6x) with respect to x. This requires applying techniques of integration for rational expressions.
Answer
$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C. $$
Answer for screen readers
The final answer to the integral is: $$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C. $$
Steps to Solve
- Simplify the integrand
The integral is given as $$ \int \frac{x^2 + 4x + 2}{6x} , dx. $$ We can simplify the expression by dividing each term in the numerator by (6x): $$ \frac{x^2}{6x} + \frac{4x}{6x} + \frac{2}{6x} = \frac{x}{6} + \frac{2}{3} + \frac{1}{3x}. $$ Thus, we rewrite the integral as: $$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{1}{3x} \right) , dx. $$
- Integrate each term separately
Now, we can integrate each term independently:
- The integral of (\frac{x}{6}) is: $$ \int \frac{x}{6} , dx = \frac{1}{6} \cdot \frac{x^2}{2} = \frac{x^2}{12}. $$
- The integral of (\frac{2}{3}) is: $$ \int \frac{2}{3} , dx = \frac{2}{3} x. $$
- The integral of (\frac{1}{3x}) is: $$ \int \frac{1}{3x} , dx = \frac{1}{3} \ln |x|. $$
- Combine the results
Combining all the results of the integrals, we have: $$ \int \frac{x^2 + 4x + 2}{6x} , dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C, $$ where (C) is the constant of integration.
The final answer to the integral is: $$ \frac{x^2}{12} + \frac{2}{3} x + \frac{1}{3} \ln |x| + C. $$
More Information
Integrating rational functions often involves simplifying the expression before applying integration rules. This particular integral demonstrated the need to treat each term independently, allowing for straightforward integration.
Tips
- Failing to simplify: Sometimes, integrals can be simplified before integrating; neglecting this can lead to increased complexity.
- Incorrect integration of logarithmic terms: Remembering the correct integral formula for ( \frac{1}{x} ).
- Forgetting the constant of integration: It's essential to always add (C) after performing an integral.
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