Which Of The Following Is Not An Improper Integral

The realm of calculus is vast and fascinating, encompassing a wide array of concepts and techniques. Among these, the concept of the improper integral holds a special place, often presenting a challenge to students. Improper integrals arise when we deal with integrals that have either infinite limits of integration or integrands that become unbounded within the interval of integration. However, not all integrals are improper.

Defining Improper Integrals

Before diving into what does not constitute an improper integral, let's clarify what an improper integral is. An integral is considered improper if it violates one or both of the following conditions:

1. Infinite Limits of Integration: At least one of the limits of integration is infinite (i.e., ∞ or −∞).
2. Unbounded Integrand: The function being integrated (the integrand) has a discontinuity within the interval of integration, meaning it approaches infinity at some point within the interval [a, b].

If an integral meets either of these criteria, it requires special techniques to evaluate, ensuring we handle the "improper" nature correctly.

What Makes an Integral "Proper"?

An integral is considered proper when it meets the following conditions:

1. Finite Limits of Integration: Both the upper and lower limits of integration are finite numbers.
2. Bounded Integrand: The function being integrated is continuous and finite over the entire interval of integration. In other words, the function does not approach infinity at any point within the interval [a, b].

If an integral satisfies both these conditions, standard integration techniques can be applied directly to find its value. No special treatment for improper behavior is needed.

Identifying Improper Integrals: Examples

To better understand the distinction, let's examine some examples of improper integrals:

1. Infinite Limit:

   ∫1∞1x2dx ​

   Here, the upper limit of integration is infinity, making it an improper integral.

2. Infinite Limit:

   ∫−∞0exdx ​

   Similarly, the lower limit is negative infinity, rendering it improper.

3. Unbounded Integrand:

   ∫011x−−√dx ​

   In this case, the integrand 1x−−√ approaches infinity as x approaches 0, which is within the interval of integration [0, 1].

4. Unbounded Integrand:

   ∫−111x2dx ​

   Here, the integrand 1x2 approaches infinity as x approaches 0, which lies within the interval [−1, 1].

Examples of Proper Integrals

Now, let's look at some examples of proper integrals:

1. Finite Limits and Bounded Integrand:

   ∫01x2dx ​

   Both limits are finite (0 and 1), and the integrand x2 is continuous and bounded on the interval [0, 1].

2. Finite Limits and Bounded Integrand:

   ∫−12sin(x)dx ​

   The limits are finite (−1 and 2), and sin(x) is continuous and bounded on the interval [−1, 2].

3. Finite Limits and Bounded Integrand:

   ∫0πcos(x)dx ​

   The limits are finite (0 and π), and cos(x) is continuous and bounded on the interval [0, π].

Which of the Following Is NOT an Improper Integral?

To answer the question directly, we need to evaluate a set of integrals and determine which one does not meet the criteria for being improper.

Let’s consider the following integrals as options:

A. ∫0∞e−xdx

B. ∫−∞∞xex2dx

C. ∫01ln(x)dx

D. ∫15x2dx

Evaluation of Each Integral:

A. ∫0∞e−xdx

• This integral has an infinite upper limit.
• Thus, it is an improper integral.

B. ∫−∞∞xex2dx

• This integral has both lower and upper limits as infinite.
• Hence, it is also an improper integral.

C. ∫01ln(x)dx

• Here, limx→0+ln(x)→−∞
• The integrand ln(x) approaches negative infinity as x approaches 0 from the positive side.
• Therefore, this is an improper integral.

D. ∫15x2dx

• The limits of integration are finite (1 and 5).
• The integrand x2 is a polynomial function, which is continuous and bounded on the entire interval [1, 5].
• Thus, this is a proper integral.

Conclusion:

The integral ∫15x2dx is not an improper integral because it has finite limits and a bounded, continuous integrand over the interval of integration.

Detailed Explanation of Proper vs. Improper Integrals

To further solidify the understanding, let's delve into the properties of proper and improper integrals with additional examples and explanations.

Properties of Proper Integrals

Proper integrals adhere to the fundamental theorem of calculus and can be evaluated using standard integration techniques. The function being integrated is well-behaved, and the interval of integration is finite. This allows for direct application of antiderivatives to find the definite integral.

Example:

Consider the integral:

∫02x3dx ​

Here, the limits of integration are 0 and 2, which are finite. The integrand x3 is a polynomial and is continuous and bounded on the interval [0, 2].

To evaluate, we find the antiderivative of x3, which is x44. Then, we apply the limits:

∫02x3dx=[x44]02=244−044=164=4 ​

Thus, the integral evaluates to 4 using standard techniques.

Properties of Improper Integrals

Improper integrals require special attention due to the infinite limits or unbounded integrands. To handle these, we use limits to approximate the integral and determine whether it converges (has a finite value) or diverges (approaches infinity).

Improper Integrals with Infinite Limits:

When dealing with infinite limits, we replace the infinite limit with a variable and take the limit as that variable approaches infinity.

Example:

Consider the integral:

∫1∞1x2dx ​

To evaluate, we replace the upper limit ∞ with a variable t and find the limit as t approaches ∞:

∫1∞1x2dx=limt→∞∫1t1x2dx

First, find the antiderivative of 1x2, which is −1x:

limt→∞∫1t1x2dx=limt→∞[−1x]1t

Now, evaluate the antiderivative at the limits:

limt→∞(−1t−(−11))=limt→∞(−1t+1)

As t approaches ∞, −1t approaches 0:

limt→∞(−1t+1)=0+1=1

Since the limit exists and is finite (1), the improper integral converges to 1.

Improper Integrals with Unbounded Integrands:

When the integrand is unbounded at a point within the interval of integration, we split the integral at that point and evaluate each part using limits.

Example:

Consider the integral:

∫011x−−√dx ​

The integrand 1x−−√ approaches infinity as x approaches 0. Thus, we replace the lower limit 0 with a variable t and take the limit as t approaches 0 from the positive side:

∫011x−−√dx=limt→0+∫t11x−−√dx

First, find the antiderivative of 1x−−√, which is 2x−−√:

limt→0+∫t11x−−√dx=limt→0+[2x−−√]t1

Now, evaluate the antiderivative at the limits:

limt→0+(21−−√−2t−−√)=limt→0+(2−2t−−√)

As t approaches 0 from the positive side, 2t−−√ approaches 0:

limt→0+(2−2t−−√)=2−0=2

Since the limit exists and is finite (2), the improper integral converges to 2.

Common Misconceptions

1. All Integrals Are Improper: Not all integrals require special techniques for evaluation. Only those with infinite limits or unbounded integrands are considered improper.
2. Improper Integrals Always Diverge: An improper integral can either converge to a finite value or diverge to infinity. Convergence depends on the specific function and limits of integration.
3. Continuity Guarantees Properness: While continuity over a closed interval [a, b] is a requirement for a proper integral, the presence of infinite limits automatically makes the integral improper, regardless of continuity.

Practical Applications

Understanding the distinction between proper and improper integrals is crucial in various fields, including:

• Physics: Calculating potential energy, work done by forces, and analyzing systems with infinite boundaries.
• Engineering: Designing structures, analyzing signals, and modeling systems with singularities.
• Statistics: Calculating probabilities in continuous distributions and determining expected values.
• Economics: Modeling economic behavior over infinite time horizons and analyzing long-term trends.

Step-by-Step Guide to Determine if an Integral is Improper

To systematically determine whether an integral is improper, follow these steps:

1. Check the Limits of Integration:
   • Are both limits finite numbers? If yes, proceed to the next step.
   • If either limit is infinite (∞ or −∞), the integral is improper.
2. Examine the Integrand:
   • Is the integrand continuous and bounded over the interval of integration? If yes, the integral is proper.
   • If the integrand has a discontinuity (approaches infinity) at any point within the interval, the integral is improper.
3. Split the Integral (if necessary):
   • If the integrand has a discontinuity at a point c within the interval [a, b], split the integral into two parts: ∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx. Then, evaluate each part separately using limits.

Examples: Applying the Step-by-Step Guide

Example 1:

∫−23x4dx

1. Limits of Integration: Both limits are finite (−2 and 3).
2. Integrand: The integrand x4 is a polynomial and is continuous and bounded on the interval [−2, 3].

Conclusion: The integral is proper.

Example 2:

∫0∞cos(x)dx

1. Limits of Integration: The upper limit is infinite (∞).

Conclusion: The integral is improper.

Example 3:

∫−111x3dx

1. Limits of Integration: Both limits are finite (−1 and 1).
2. Integrand: The integrand 1x3 has a discontinuity at x = 0, which is within the interval [−1, 1].

Conclusion: The integral is improper.

Advanced Techniques for Evaluating Improper Integrals

While evaluating improper integrals using limits is the fundamental approach, there are more advanced techniques that can simplify the process in certain cases. Some of these include:

1. Comparison Test: Used to determine the convergence or divergence of an improper integral by comparing it to another integral whose convergence is known.
2. Limit Comparison Test: Similar to the comparison test but involves taking the limit of the ratio of the integrands to determine convergence or divergence.
3. Integration by Parts: Useful for integrals involving products of functions, especially when one of the functions simplifies upon differentiation.
4. Residue Theorem (Complex Analysis): A powerful technique for evaluating certain types of improper integrals using complex analysis.

Conclusion

In summary, understanding the distinction between proper and improper integrals is fundamental in calculus. An integral is considered proper if it has finite limits of integration and a bounded, continuous integrand over that interval. Conversely, an integral is improper if it has infinite limits or an unbounded integrand within the interval. The integral ∫15x2dx is a quintessential example of a proper integral, as it meets both criteria: finite limits and a well-behaved integrand. By systematically checking these conditions, one can accurately classify integrals and apply the appropriate evaluation techniques.