How To Integrate X 2 1 X 2

How to Integrate x²/(1+x²)

This article will guide you through the process of integrating the function x²/(1+x²). This integral is a classic example that combines several integration techniques and provides a great illustration of how to approach complex integration problems. Understanding this process will enhance your calculus skills significantly. We'll explore different methods and demonstrate how to arrive at the solution effectively.

Understanding the Integrand

The integrand, x²/(1+x²), presents a challenge because we can't directly apply a simple power rule. It requires a bit of manipulation to make it integrable. The key here is recognizing that we can rewrite the integrand using polynomial long division or a clever algebraic trick.

Method 1: Polynomial Long Division

The most straightforward approach is to use polynomial long division to simplify the integrand. Dividing x² by (1+x²) gives:

x² / (1+x²) = 1 - 1/(1+x²)

Now, the integral becomes significantly easier:

∫x²/(1+x²) dx = ∫[1 - 1/(1+x²)] dx

This integral can now be solved using basic integration rules:

∫1 dx = x + C₁

and

∫1/(1+x²) dx = arctan(x) + C₂

Therefore, the final solution using this method is:

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

Method 2: Algebraic Manipulation

Alternatively, we can manipulate the integrand algebraically. Add and subtract 1 from the numerator:

x²/(1+x²) = (x² + 1 - 1) / (1+x²) = (1+x²)/(1+x²) - 1/(1+x²) = 1 - 1/(1+x²)

This leads us to the same integral as in Method 1, resulting in the identical solution:

∫x²/(1+x²) dx = x - arctan(x) + C

Understanding the Solution

The solution, x - arctan(x) + C, might seem unexpected. However, it's perfectly valid. The arctangent function, arctan(x), represents the inverse tangent function, giving the angle whose tangent is x. The constant of integration, C, accounts for the fact that there are infinitely many antiderivatives, all differing by a constant.

Verifying the Solution (Differentiation)

To verify our solution, we can differentiate it and see if we get back the original integrand:

d/dx [x - arctan(x) + C] = 1 - 1/(1+x²) = (1+x² - 1)/(1+x²) = x²/(1+x²)

The differentiation confirms that our solution is correct.

Conclusion

Integrating x²/(1+x²) successfully involves simplifying the integrand using polynomial long division or algebraic manipulation. This allows us to apply basic integration rules to obtain the final solution, x - arctan(x) + C. Remember to always verify your solutions by differentiating the result. This problem showcases the importance of algebraic manipulation and strategic application of integration techniques in solving seemingly complex integrals. Mastering these techniques will prove invaluable in your further studies of calculus.