Problem 7 - Calculus of a Single Variable

Question

Determining Whether an Integral Is Improper In Exercises \(1-8\) , decide whether the integral is improper. Explain your reasoning. $$ \int_{-\infty}^{\infty} \frac{\sin x}{4+x^{2}} d x $$

Step-by-Step Solution

Answer

Yes, the given integral is an improper integral due to its infinite limits of integration.

Step 1: Identifying Integral Type

Firstly, observe the given integral: \(\int_{-\infty}^{\infty} \frac{\sin x}{4+x^{2}} dx\). Here, the limits of integration are \(-\infty\) and \(+\infty\), which are infinite limits. This indicates that it is an improper integral.

Step 2: Checking for Singularity

Next, check if the integrand function \(\frac{\sin x}{4+x^{2}}\) has any singularities within the interval of integration. The denominator of the integrand is \(4+x^{2}\). So, look for the values of \(x\) at which the denominator is zero. The equation \(4+x^{2}=0\) has no real roots. Hence, there are no singularities within the interval of integration.

Step 3: Concluding

Therefore, the integral \(\int_{-\infty}^{\infty} \frac{\sin x}{4+x^{2}} dx\) is an improper integral due to its infinite limits of integration, even though it does not have any singularities.

Key Concepts

Infinite Limits

Integrals with infinite limits, like the one in this problem, often appear daunting at first. However, they are simply a type of improper integral that extends to either infinity or negative infinity. These integrals require unique approaches because they span an unbounded region on the number line.
Understanding how to work with infinite limits can drastically simplify tackling integrals of this kind.

• When you see \(-\infty\) or \(\infty\) as the limits of integration, you are dealing with an improper integral.
• Proper methods to compute these limits are vital as the integral depends on these values being managed correctly.

Essentially, the behavior of the integrand function becomes more crucial since the function must remain manageable even as the integration applies over an infinite range.By breaking down into limits, each piece can often be evaluated more easily, providing insights into how the function behaves over time.

Singularities

Singularities within integrals occur when the integrand function becomes undefined or diverges at a point within the limits of integration. In this exercise, the focus is on understanding if there are any singular points.
Determining the presence of singularities involves evaluating where the denominator of the integrand takes values that disrupt calculation:

• For \(\frac{\sin x}{4+x^2}\), the denominator is \(4+x^2\).
• Setting the denominator equal to zero without yielding real roots confirmed no singularities existed for this scenario.

Conclusively, checking for singularities is crucial since they can significantly impact the integral's value and the methods used to find it.

Integrand Function

The integrand function is the expression being integrated, which in this problem is \(\frac{\sin x}{4+x^2}\). This function determines much about the integral's properties, including how you handle improper integrals.
The characteristics of the integrand function include:

• The numerator \(\sin x\) indicates oscillatory behavior, common in trigonometric functions.
• The denominator \(4+x^2\) ensures the function does not become undefined, due to lack of real roots.

Together, these characteristics allow the integrand function to be continuous and integrable over the entire real line, further analyzing the function aids in setting strategies for calculus problem-solving.

Calculus Problem Solving

Solving calculus problems, particularly with improper integrals, often requires a strategy that systematically addresses each component involved in the problem. This problem-solving approach focuses on:

• Identifying integral types: Start by noting any infinite limits or potential singularities.
• Checking results step-by-step: Ensure thorough examination of the function and interval limits.
• Safeguarding calculations: Remember to address improper integrals with definite techniques, like transforming limits when necessary.

By working through these steps carefully, students can develop a deeper understanding of the underlying calculus principles. Each calculus problem presents an opportunity for applying learned techniques and gaining new insights, reinforcing fundamental calculus concepts.