∫ (x² + 4x + 7) / (6x) dx
Understand the Problem
The question involves evaluating the integral of a polynomial function over a variable x. The expression to integrate is given as a fraction, which needs to be simplified and then integrated accordingly.
Answer
$$ \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln |x| + C $$
Answer for screen readers
The integral is given by: $$ \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln |x| + C $$
Steps to Solve
- Simplify the Fraction
To simplify the expression (\frac{x^2 + 4x + 7}{6x}), we can separate it into individual fractions: $$ \int \left( \frac{x^2}{6x} + \frac{4x}{6x} + \frac{7}{6x} \right)dx $$ This simplifies to: $$ \int \left( \frac{x}{6} + \frac{2}{3} + \frac{7}{6x} \right)dx $$
- Integrate Each Term
Now, we 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{7}{6x}): $$ \int \frac{7}{6x} , dx = \frac{7}{6} \ln |x| $$
- Combine the Results
Now, we combine the results of the integrations: $$ \int \left( \frac{x^2 + 4x + 7}{6x} \right)dx = \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln |x| + C $$ where (C) is the constant of integration.
The integral is given by: $$ \frac{x^2}{12} + \frac{2}{3} x + \frac{7}{6} \ln |x| + C $$
More Information
This integral demonstrates the use of basic integration techniques, including separating fractions and integrating polynomial functions and logarithms. The result includes a logarithmic term, common in integrals involving rational functions.
Tips
- Forgetting the constant of integration: Always remember to include the constant (C) when performing indefinite integrals.
- Mistaking terms during separation: Make sure each term of the fraction is correctly simplified before integrating.
