Problem 9 - Calculus for Biology and Medicine

Question

All the integrals are improper and converge. Explain in each case why the integral is improper, andevaluate each integral. $$ \int_{-\infty}^{\infty} \frac{x}{\left(1+x^{2}\right)^{2}} d x $$

Step-by-Step Solution

Answer

The integral is improper due to infinite limits and converges to 0 due to symmetry of the integrand.

Step 1: Identify Improper Nature

The integral \( \int_{-\infty}^{\infty} \frac{x}{(1+x^2)^2} \, dx \) is improper due to its infinite limits of integration, extending from \(-\infty\) to \(\infty\). This requires us to evaluate the integral using limits to handle the infinite bounds.

Step 2: Set up the Limit for Improper Integral

We express the integral as a limit: \[ \lim_{a \to -\infty, \, b \to \infty} \int_{a}^{b} \frac{x}{(1+x^2)^2} \, dx. \] This breaks the problem into a manageable form, allowing us to evaluate finite bounds first, then take the limits.

Step 3: Check the Function's Symmetry

Observe that the integrand, \( \frac{x}{(1+x^2)^2} \), is an odd function (since \( f(-x) = -f(x) \)). When integrating an odd function symmetrically around \(0\) over an interval from \(-c\) to \(c\), the result is zero due to symmetry.

Step 4: Confirm Convergence

Despite the infinite interval, the symmetry property simplifies evaluation. For any interval from \(-c\) to \(c\), since the integrand is odd, the integral evaluates to zero. Therefore, as \(a \to -\infty\) and \(b \to \infty\), the integral remains zero, confirming convergence.

Step 5: Evaluate the Final Result

Since the improper integral evaluates as a limit of symmetric bounds of an odd function around the origin, the result is \[ \int_{-\infty}^{\infty} \frac{x}{(1+x^2)^2} \, dx = 0. \]

Step 6: Conclusion

By using symmetry properties for an odd function, we determine that the given improper integral converges to zero.

Key Concepts

Infinite Limits

Improper integrals often have infinite limits. These are a key part of identifying why an integral might be considered improper. When we talk about infinite limits, we are referring to integrals that extend to infinity, either positively or negatively.
In the problem at hand, the integration is performed over the interval from \(-\infty\) to \(\infty\). This means that the function \(\frac{x}{(1+x^2)^2}\) is being integrated over an infinitely large region on the x-axis.
Handling these requires a specific strategy: converting the problem into a limit problem. The goal is to confine calculations to a smaller segment and later allow this segment to stretch infinitely through limits.

Limit Evaluation

Taking a limit evaluation transforms an improper integral into a problem with finite bounds by using limits. This involves breaking the integral into sections where one limit approaches \( -\infty \) and the other approaches \( \infty \.\)
To evaluate our integral, we express it as \[ \lim_{a \to -\infty, \, b \to \infty} \int_{a}^{b} \frac{x}{(1+x^2)^2} \, dx. \]
Applying limits in this way allows students to deal with an initially endless problem in easier, manageable steps. We first evaluate the definite integral from \(a\) to \(b\), then seek out their behavior as they stretch toward infinity. This step-wise process is both effective and essential to ensure accurate evaluation.

Symmetry Properties

Symmetry properties, especially when dealing with odd functions or symmetrical intervals, can simplify the evaluation of integrals. An odd function satisfies \(f(-x) = -f(x)\).
In this exercise, the integrand function \(\frac{x}{(1+x^2)^2}\) is identified as an odd function. This symmetry assists us greatly when the integration bounds are symmetric around zero:

• An integration from \(-c\) to \(c\) of an odd function results in zero.

Knowing this, we quickly determine that, for any symmetric bounds, the contributions from \(x\) and \(-x\) cancel out each other. Hence, the result is zero. This simplifies solving the integral tremendously when dealing with infinite bounds.

Convergent Integrals

For an improper integral to be useful or meaningful, it must converge to a specific value. Convergent integrals settle into a finite limit, indicating a recognizable total accumulation.
In this exercise, despite the potentially infinite span between the bounds \(-\infty\) to \(\infty\), the symmetry of the function ensures that the total integral converges to zero. Since the parts cancel each other out perfectly due to symmetry, the combined area totaled by the integral is exactly zero, denoting convergence.
Convergence is crucial in mathematics because it tells us that the value of the integral is not indefinite but rather a finite and precise number. Recognizing convergence helps us trust the process and results derived from improper integrals.