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6.13 Evaluating Improper Integrals

3 min readβ€’january 29, 2023


AP Calculus AB/BC ♾️

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Evaluating Improper Integrals

Improper Integrals are integrals where either one or both of the limits of integration are infinite or the integrand has a singularity in the interval of integration. AP Calculus BC students should know how to evaluate such integrals.
Here are the steps to evaluate improper integrals:
  1. Identify the type of improper integral: Improper integrals can be either improper definite or improper indefinite integrals. Improper definite integrals have both limits of integration as either infinite or finite, while improper indefinite integrals have either one or both limits of integration as infinite.
  2. Identify the singularity or infinite limit: The next step is to identify the singularity or infinite limit of the integrand. This is crucial because the method for evaluating the improper integral will depend on the type of singularity or infinite limit.
  3. Use one of the following techniques to evaluate the improper integral: a. Constraining the interval of integration: Constraining the interval of integration means breaking the integral into smaller pieces and evaluating the integral for each piece separately. This technique is useful for integrands that have a singularity in the interval of integration. b. Comparison test: The comparison test is used to determine if the integral converges or diverges by comparing the integrand with a function that is known to converge or diverge. c. Integrating by parts: Integrating by parts is used to evaluate integrals with infinite limits of integration. The method involves finding an antiderivative of the integrand and applying L'HΓ΄pital's Rule. d. Trigonometric substitution: Trigonometric substitution is used to evaluate integrals that involve square roots of polynomials. This technique involves changing the integrand into a function that can be integrated using trigonometry. e. Improper Integral tables: Improper integral tables are useful for evaluating integrals that cannot be done using the above techniques. The table contains a list of integrals with their corresponding solutions.

Example Problems:

Example 1: Consider the improper definite integral ∫_0^∞ (e^(-x^2)) dx. To evaluate this integral, use the comparison test. By comparing the integrand with a known convergent function, we find that the integral converges to the value of βˆšΟ€.
Example 2: Consider the improper definite integral ∫_1^∞ (x^(-2)) dx. To evaluate this integral, use the technique of constraining the interval of integration. By breaking the integral into two pieces and evaluating the integral for each piece, we find that the integral diverges to ∞.
Example 3: Consider the improper indefinite integral ∫ (e^x / (1 + e^x)) dx. To evaluate this integral, use integrating by parts. By finding an antiderivative of the integrand and applying L'Hôpital's Rule, we find that the integral evaluates to ln|(1 + e^x)| + C, where C is a constant of integration.
Example 4: Consider the improper definite integral ∫_0^1 (1/√(x)) dx. To evaluate this integral, use the comparison test. By comparing the integrand with a known convergent function, we find that the integral converges to 2.
Example 5: Consider the improper definite integral ∫_1^∞ (1/x^(3/2)) dx. To evaluate this integral, use the technique of constraining the interval of integration. By breaking the integral into two pieces and evaluating the integral for each piece, we find that the integral converges to 2√(2).
Example 6: Consider the improper indefinite integral ∫ (1/√(x-1)) dx. To evaluate this integral, use trigonometric substitution. By changing the integrand into a function that can be integrated using trigonometry, we find that the integral evaluates to 2√(x-1) + C, where C is a constant of integration.
In conclusion, evaluating improper integrals is a crucial skill for AP Calculus BC students. To evaluate such integrals, students should be familiar with the various techniques such as constraining the interval of integration, comparison test, integrating by parts, trigonometric substitution, and improper integral tables. By following these steps and techniques, students will be able to evaluate any type of improper integral, whether it be a definite or indefinite integral, and whether it has a singularity or infinite limit in the interval of integration. As improper integrals are often seen in real-world applications, mastering this topic will help students to better understand and analyze these applications.
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