Question

Evaluate the integrals. If the integral diverges, answer "diverges."\(\int_{-\infty}^{\infty} \frac{x}{\left(x^{2}+1\right)^{2}} d x\)

Short answer

The integral evaluates to 0.

Step-by-step solution

Consider the Integral as an Improper Integral

The given integral is \[\int_{-\infty}^{\infty} \frac{x}{(x^2+1)^2} \ dx.\]This is an improper integral because it integrates over an infinite interval, from \(-\infty\) to \(\infty\). We need to evaluate it as a limit.

Split the Integral into Two Parts

We can split the integral into two parts at 0, since it's an even range around the y-axis:\[\int_{-\infty}^{0} \frac{x}{(x^2+1)^2} \ dx + \int_{0}^{\infty} \frac{x}{(x^2+1)^2} \ dx.\]

Analyze the Function's Symmetry

The function \(\frac{x}{(x^2+1)^2}\) is an odd function because if you replace \(x\) with \(-x\), you get:\[\frac{-x}{((-x)^2+1)^2} = -\frac{x}{(x^2+1)^2}.\]Since the function is odd, the integral over any symmetric interval around the y-axis will be zero.

Conclude the Result of the Integral

Because the integrand is an odd function and the limits are symmetric about the origin (from \(-\infty\) to \(\infty\)), the integral evaluates to 0.

Key concepts

Odd Functions

When tackling an integral, especially over symmetric intervals like from \(-\infty\) to \(\infty\), identifying the function's symmetry can save a lot of work. An odd function, in mathematical terms, is a function that satisfies the condition \(f(-x) = -f(x)\) for every point x in its domain. This means that the graph of the function is symmetric with respect to the origin.

For example, if you think about the function \(f(x) = \frac{x}{(x^2+1)^2}\), replacing \(x\) with \(-x\) gives us: \(-\frac{x}{(x^2+1)^2}\), which is clearly the negative of the original function. Thus, confirming it is indeed an odd function.

One interesting property of odd functions is that when integrated over a symmetric interval around the origin, the integral will always result in zero. This occurs because the negative areas on one side of the y-axis will perfectly cancel out the positive areas on the other side. In our exercise, where the integral is evaluated from \(-\infty\) to \(\infty\), this principle simplifies the evaluation process considerably.

Infinite Interval Integration

Improper integrals, such as those that span from \(-\infty\) to \(\infty\), involve integrating over an infinite interval. This requires a careful approach since we are dealing with limits rather than finite endpoints. In simple terms, we can't just "plug in" infinity, which means we need to approach this from a limit perspective.

To evaluate such integrals, you generally split the integral into two parts along a convenient point, often zero for symmetric functions. Using the notation of limits, you can write:

• \(\int_{-\infty}^{\infty} f(x) \, dx = \lim_{{a \to -\infty}} \int_{a}^{0} f(x) \, dx + \lim_{{b \to \infty}} \int_{0}^{b} f(x) \, dx\)

This helps in managing the behavior of the function as it approaches infinity in both directions.
Splitting simplifies dealing with infinities one side at a time, allowing us to analyze how each side converges or behaves independently.

Symmetry in Integration

Utilizing symmetry can be a powerful tool when evaluating integrals. If a given function demonstrates symmetry, particularly odd symmetry, and the boundaries of integration themselves reflect this symmetry, it can directly inform the result of the integration.

In our exercise, the function \(\frac{x}{(x^2+1)^2}\) displayed odd symmetry and was integrated over a symmetric interval around the origin: from \(-\infty\) to \(\infty\). This symmetric setup means that whatever area lies above the x-axis on one side of the origin, an equivalent area lies below on the opposite side, effectively cancelling out any net area.

Thus, symmetry allows for determining such integrals quickly without performing detailed calculations, leveraging the geometric properties of the function. By graphically visualizing the function and its intervals, you can often predict the zero result of integration, facilitating a deeper understanding and quicker problem-solving.