Domanda del libro di testo
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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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.PE.59
Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to ∞ of (x² * e^(−x)) dx
Guida verificata passo dopo passo1
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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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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Percorso guidato
11:11
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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Percorso guidato
06:18Integration 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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5:22Factorials
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Domanda del libro di testo
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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Domanda del libro di testo
Evaluate the integrals in Exercises 1–8 using integration by parts.
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Domanda del libro di testo
135. Evaluate ∫₀^(π/2) (sin x) / (sin x + cos x) dx in two ways:
(a) By evaluating ∫ (sin x) / (sin x + cos x) dx, then using the Evaluation Theorem.
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Domanda del libro di testo
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
129. ∫ (x^(ln x) * ln x) / x dx
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Domanda del libro di testo
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (2 − cosx + sinx) / sin²x dx
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