How To Integrate X 2 X 2 1

How to Integrate x²/(x² + 1) dx: A Step-by-Step Guide

This article will guide you through the process of integrating the function x²/(x² + 1) with respect to x. This seemingly simple integral requires a clever approach, utilizing polynomial long division or a strategic algebraic manipulation. We'll explore both methods, ensuring you understand the underlying principles and can tackle similar integration problems.

Understanding the Problem: The integral ∫x²/(x² + 1) dx isn't immediately solvable using basic integration rules. We need to manipulate the integrand to make it more manageable. The core concept revolves around simplifying the fraction.

Method 1: Polynomial Long Division

This method involves dividing the numerator (x²) by the denominator (x² + 1).

1. Perform Long Division:

       1
   x² + 1 | x² + 0x + 0
           -x² - 1
           -------
               -1 
   

2. Rewrite the Integrand: The result of the long division shows that x²/(x² + 1) = 1 - 1/(x² + 1).

3. Integrate Term by Term: Now we can rewrite the integral as: ∫[1 - 1/(x² + 1)] dx

4. Solve the Integral: This integral breaks down into two simpler integrals: ∫1 dx - ∫1/(x² + 1) dx

   The integral of 1 with respect to x is simply x. The integral of 1/(x² + 1) is arctan(x) (or tan⁻¹(x)).

5. Final Result: Therefore, the final solution is: x - arctan(x) + C (where C is the constant of integration).

Method 2: Algebraic Manipulation

This method cleverly manipulates the integrand without explicitly performing long division.

1. Add and Subtract 1 in the Numerator: We can rewrite the numerator as (x² + 1) - 1.

2. Separate the Fractions: This allows us to split the integrand into two separate fractions: ∫[(x² + 1)/(x² + 1) - 1/(x² + 1)] dx

3. Simplify: The first fraction simplifies to 1, leaving us with: ∫[1 - 1/(x² + 1)] dx

4. Solve the Integral (Same as Method 1): From this point, the solution follows the same steps as in Method 1, leading to the same final result: x - arctan(x) + C

Conclusion:

Both methods demonstrate effective strategies for solving this integral. The choice between polynomial long division and algebraic manipulation often comes down to personal preference and the specific problem's structure. Understanding both techniques broadens your problem-solving capabilities in calculus. Remember to always include the constant of integration (C) in your final answer, as this accounts for the family of antiderivatives. This integral serves as an excellent example of how manipulating the integrand can simplify complex integration problems. Practicing both methods will strengthen your understanding of integration techniques and prepare you for more challenging integrals in the future.