Integral Of X 2 X 2 1

Solving the Integral of x²/(x²+1)

This article will guide you through the process of solving the indefinite integral of x²/(x²+1). This integral might seem daunting at first, but with the right approach, it becomes quite manageable. We'll explore a straightforward method using polynomial long division and then briefly touch upon an alternative approach using trigonometric substitution. This approach is ideal for beginners and demonstrates a fundamental technique applicable to a range of integration problems.

Understanding the Problem:

We aim to find the indefinite integral: ∫x²/(x²+1) dx. This integral is not immediately solvable using basic integration rules. However, we can simplify the integrand using polynomial long division.

Method 1: Polynomial Long Division

The first step is to perform polynomial long division to simplify the integrand. Dividing x² by (x²+1), we get:

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

This division reveals that x²/(x²+1) = 1 - 1/(x²+1). Substituting this back into our integral, we obtain:

∫x²/(x²+1) dx = ∫[1 - 1/(x²+1)] dx

Now, we can integrate term by term:

∫1 dx = x + C₁ (where C₁ is the constant of integration)

∫1/(x²+1) dx = arctan(x) + C₂ (where C₂ is the constant of integration)

Therefore, the solution to the integral is:

∫x²/(x²+1) dx = x - arctan(x) + C (where C = C₁ + C₂ is the constant of integration)

Method 2: Trigonometric Substitution (Brief Overview)

While polynomial long division is the more straightforward approach for this particular integral, it's worth mentioning that trigonometric substitution is another viable method. This involves substituting x = tan(θ), which then transforms the integral into a trigonometric integral solvable using trigonometric identities. This method can be more complex and is generally preferred for integrals involving expressions like √(a² - x²), √(a² + x²), or √(x² - a²). For this specific problem, polynomial long division is significantly simpler.

Key Points to Remember:

• Polynomial long division: A powerful technique for simplifying rational functions before integration.
• Term-by-term integration: Allows us to break down complex integrals into simpler, manageable parts.
• Constants of integration: Always remember to include the constant of integration (+C) in indefinite integrals.
• Choosing the right method: Consider the structure of the integrand when deciding on the best integration technique.

Conclusion:

This article demonstrated a clear and concise method for solving the integral of x²/(x²+1) dx. By using polynomial long division, we effectively simplified the integrand, enabling us to easily integrate each term. While alternative methods exist, such as trigonometric substitution, the approach detailed here provides a straightforward and efficient solution for this particular problem. Understanding these techniques will equip you to tackle similar integration problems with confidence.