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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.PE.59
Chapter 8, Problem 8.PE.59

Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to ∞ of (x² * e^(−x)) dx

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Recognize that the integral \( \int_0^{\infty} x^2 e^{-x} \, dx \) is an improper integral because the upper limit is infinity. This means we need to evaluate it as a limit: \( \lim_{t \to \infty} \int_0^t x^2 e^{-x} \, dx \).
Set up the integral with the limit: \( \lim_{t \to \infty} \int_0^t x^2 e^{-x} \, dx \). We will first find the antiderivative of \( x^2 e^{-x} \) and then apply the limits.
Use integration by parts to find the antiderivative. Let \( u = x^2 \) and \( dv = e^{-x} dx \). Then, compute \( du = 2x dx \) and \( v = -e^{-x} \). Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \).
After the first integration by parts, you will get an integral involving \( x e^{-x} \). Apply integration by parts again on this new integral, letting \( u = x \) and \( dv = e^{-x} dx \), and repeat the process to fully evaluate the integral.
Once you have the antiderivative, evaluate it at the limits 0 and \( t \), then take the limit as \( t \to \infty \). This will give you the value of the improper integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, the integral is expressed as a limit where the bound approaches infinity or the point of discontinuity. Convergence or divergence is determined by the existence of this limit.
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Improper Integrals: Infinite Intervals

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation, used to integrate products of functions. It transforms the integral of u dv into uv minus the integral of v du, simplifying complex integrals like x² e^(−x). Choosing u and dv wisely is key to success.
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Integration by Parts for Definite Integrals

Gamma Function and Factorials

The Gamma function generalizes factorials to non-integer values and is defined as an improper integral involving x^(n) e^(−x). Recognizing integrals of the form ∫₀^∞ x^n e^(−x) dx as Gamma functions helps evaluate them quickly, where Γ(n+1) = n! for positive integers n.
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Factorials
Related Practice
Textbook Question

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [cos(θ) / (sin²(θ) + sin(θ) − 6)] dθ

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Textbook Question

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

125. ∫ dx / (√x * √(1 + x))

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Evaluate the integrals in Exercises 1–8 using integration by parts.

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(a) By evaluating ∫ (sin x) / (sin x + cos x) dx, then using the Evaluation Theorem.

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Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

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Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

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