Understanding the Integral
To solve the integral of (2x² + 2x + 3)/(x⁴ + 2x³ + 3x² + 2x + 2), we will break it down into manageable steps.
Step 1: Simplifying the Denominator
- Identify the polynomial in the denominator: x⁴ + 2x³ + 3x² + 2x + 2.
- Check for possible factorization or roots to simplify the expression.
Step 2: Polynomial Long Division
- Since the degree of the numerator is less than the denominator, we can perform polynomial long division if necessary.
- If the numerator were of equal or higher degree, this would be essential for simplification.
Step 3: Partial Fraction Decomposition
- If the denominator factors nicely, consider using partial fractions.
- Break down the fraction into simpler fractions that are easier to integrate.
Step 4: Integration Techniques
- Depending on the result from partial fractions, apply appropriate integration techniques such as substitution or direct integration.
- Common techniques include integration by parts or trigonometric substitution if applicable.
Step 5: Final Integration
- Integrate each term obtained from partial fractions or long division.
- Combine the results and add the constant of integration (C) at the end.
Conclusion
- The final result will give you the integral of the original expression.
- Verify your result by differentiating back to the original function.
This methodical approach will help you efficiently tackle similar problems in the JEE category. For more practice and detailed explanations, consider exploring resources available on EduRev.