Problem 39 - Calculus Single Variable

Question

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{\infty} \frac{1}{4+x^{2}} d x $$

Step-by-Step Solution

Answer

The integral converges, and its value is \( \frac{\pi}{2} \).

Step 1: Recognize the Type of Integral

The given integral \( \int_{-\infty}^{\infty} \frac{1}{4+x^{2}} \, dx \) is an improper integral due to the infinite limits of integration. This type requires checking for convergence and, if it converges, calculating the integral's value.

Step 2: Break the Integral into Two Parts

Split the integral at a convenient point, such as zero, to handle infinite limits separately: \( \int_{-\infty}^{0} \frac{1}{4+x^{2}} \, dx + \int_{0}^{\infty} \frac{1}{4+x^{2}} \, dx \). Each part will be treated as an improper integral with one infinite limit.

Step 3: Use Substitution for Evaluation

Recognize the integral resembles the arctangent function. Recall \( \int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \arctan \left(\frac{x}{a}\right) + C \). Here, \( a = 2 \), so transform both parts: \( \lim_{b \to -\infty} \int_{b}^{0} \frac{1}{4+x^{2}} \, dx + \lim_{c \to \infty} \int_{0}^{c} \frac{1}{4+x^{2}} \, dx \).

Step 4: Evaluate Each Half

For the first part, using substitution: \( \int \frac{1}{4+x^{2}} \, dx = \frac{1}{2} \arctan \left(\frac{x}{2}\right) \). Calculate: \( \lim_{b \to -\infty} \left[ \frac{1}{2} \arctan \left(\frac{x}{2}\right) \right]_{b}^{0} = \frac{1}{2} \left( \arctan(0) - \arctan\left(\frac{b}{2}\right) \right) \). As \( b \to -\infty \), \( \arctan\left(\frac{b}{2}\right) \to -\frac{\pi}{2} \), so this half simplifies to \( \frac{\pi}{4} \).

Step 5: Evaluate the Second Half

Similarly, for the second part: \( \lim_{c \to \infty} \left[ \frac{1}{2} \arctan \left(\frac{x}{2}\right) \right]_{0}^{c} = \frac{1}{2} \left( \arctan\left(\frac{c}{2}\right) - \arctan(0) \right) \). As \( c \to \infty \), \( \arctan\left(\frac{c}{2}\right) \to \frac{\pi}{2} \), giving a value of \( \frac{\pi}{4} \).

Step 6: Combine Results

Add the results of both integrals to determine convergence: \( \frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2} \). Hence, the integral converges.

Key Concepts

Convergence of Integrals

In calculus, the concept of convergence is crucial when dealing with improper integrals. An improper integral, such as the one given in this exercise, has infinite or undefined limits of integration. To determine if such an integral converges, we look at the behavior of the function as it approaches these limits.

• If the integral approaches a finite number, it is said to converge.
• If it approaches infinity (or negative infinity), it diverges.

For the integral in this exercise, \( \int_{-\infty}^{\infty} \frac{1}{4+x^{2}} \, dx \), the idea is to check if its value is finite as the limits of integration approach infinity and negative infinity. The convergence tells us that the "area under the curve" described by this function is finite. Hence, this integral converges if we find its total accumulated area is a specific number.

Arctangent Function

The arctangent function, denoted as \( \arctan(x) \), is the inverse of the tangent function. Its significance in evaluating integrals comes from its appearance in the integral of the form \( \int \frac{1}{a^2 + x^2} \, dx \). This integral evaluates to \( \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C \), where \( C \) is the constant of integration.

• This result helps us solve integrals involving simple rational functions.
• For the integral \( \int \frac{1}{4+x^{2}} \, dx \), notice that it's rewritten in the form utilizing \( a = 2 \).

This makes the evaluation straightforward, as we directly use the antiderivative given by the arctangent formula. Additionally, the arctangent function has a range from \(-\frac{\pi}{2}\) to \(+\frac{\pi}{2}\), ensuring that the final evaluated value stays within this finite range as part of the convergence criteria.

Integral Evaluation Steps

When approaching an integral with infinite limits, it's helpful to break it down into manageable parts. In this exercise, the integral is split into two, allowing each to be evaluated separately:

• \( \int_{-\infty}^{0} \frac{1}{4+x^{2}} \, dx \)
• \( \int_{0}^{\infty} \frac{1}{4+x^{2}} \, dx \)

Each part becomes an improper integral with one infinite bound.

• We then use the known result for the arctangent function to evaluate each half.
• The limits are taken into account: as \( x \) approaches infinity or negative infinity, the arctangent applies its respective limits, either \(-\frac{\pi}{2}\) or \(\frac{\pi}{2}\).

Upon evaluating these two parts, the results are combined for the final answer. This step-by-step breakdown ensures an accurate assessment of convergence and allows for a focused approach to evaluating improper integrals.

Infinite Limits of Integration

Handling infinite limits of integration requires a procedural approach to ensure accurate evaluation. In this exercise, the integral extends from \(-\infty\) to \(\infty\), which categorizes it as an improper integral.

• To tackle this infinity, integrals like these are approached systematically.
• By splitting the integral at zero, we can focus on one infinite boundary at a time.

With each split integral, limiting processes are applied:

• For the limits approaching negative or positive infinity, substitutions or transformations help manage these boundaries.
• The arctangent function's properties, particularly as \( x \) tends toward infinity, guide the final evaluation.

Ultimately, the sum of the evaluated integrals results in converging to a specific number, demonstrating that the integral from \(-\infty\) to \(\infty\) is indeed possible to evaluate accurately despite its infinite nature.