Problem 13 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(1-34\) without using tables. $$ \int_{-\infty}^{\infty} \frac{2 x d x}{\left(x^{2}+1\right)^{2}} $$

Step-by-Step Solution

Answer

The integral evaluates to 0 due to symmetry of the odd function.

Step 1: Identify the Type of Integral

The given integral is \( \int_{-\infty}^{\infty} \frac{2x \, dx}{(x^2+1)^2} \). It is a definite integral from \(-\infty\) to \(\infty\) of a rational function. Because of the infinite limits, we need to utilize techniques suitable for improper integrals.

Step 2: Check Symmetry

The integrand \(f(x) = \frac{2x}{(x^2+1)^2}\) is an odd function because \(f(-x) = -f(x)\). For an odd function integrated symmetrically about zero, the result is zero because the area on the right cancels the area on the left.

Step 3: Evaluate the Integral (Symmetry Argument)

Since the function is odd and the limits are symmetric about zero (\(-\infty\) to \(\infty\)), the integral evaluates to zero. Hence, \[\int_{-\infty}^{\infty} \frac{2x \, dx}{(x^2+1)^2} = 0.\]

Key Concepts

Definite Integral

A definite integral gives us the area under a curve between two specific limits. When we talk about integrals, they are mostly of this type where the limits define the interval over which we want to calculate the area. A definite integral is represented as \(\int_{a}^{b} f(x) \, dx\) where \(a\) and \(b\) are the limits. The definite integral results in a numerical value, representing the net area.

• It accounts for areas above the x-axis as positive and below the x-axis as negative.
• Final result is the net area, found by subtracting the negative area from the positive area.

Understanding definite integrals is crucial in calculus because they help quantify the total accumulation of quantities, be it distance, area, or any physical quantity over time.

Improper Integral

Improper integrals extend the concept of definite integrals to cases where one or both limits of integration are infinite or the function has an infinite discontinuity. In our original exercise, the integrals go from \\(-\infty\) to \(\infty\), making it an improper integral.

• These types of integrals are crucial in dealing with functions that don't fit within the typical bounded range.
• When evaluating, it's common to take the limit as the boundary approaches infinity to see if the integral converges to a finite number.

Handling improper integrals requires careful analysis and techniques such as limit evaluation, which were used in the solution when observing the integrand symmetry.

Rational Function

Rational functions are ratios of two polynomials, which are extremely crucial in calculus due to their appearance in many real-world situations and mathematical models. A typical rational function looks like \( \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials.

• In calculus, these functions are often subject to techniques such as partial fraction decomposition to simplify integration.
• They can sometimes be improper (when the degree of the numerator is greater than or equal to the degree of the denominator), requiring extra steps for simplification.

In our problem, the function \( \frac{2x}{(x^2+1)^2} \) is a rational function because both numerator and denominator are polynomials.

Symmetry in Integrals

Symmetry is a powerful tool in calculus as it can greatly simplify the evaluation of integrals. The function \( f(x) \) is said to be symmetric if it exhibits certain repetitive features when reflected in either the x-axis or y-axis. In our example, the integrand \( \frac{2x}{(x^2+1)^2} \) is an odd function. We can affirm this because \( f(-x) = -f(x) \).

• Odd functions have a unique property where the integral from \(-a\) to \(a\) yields zero, due to area on one side of the y-axis canceling the other.
• Exploiting symmetry, especially for even or odd functions, saves significant calculations by possibly immediately determining the result.

Hence, understanding the symmetry in integrals not only simplifies the computation but also provides deep insights into the behavior of the function involved.