Domanda del libro di testo
Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - cos²(θ)) 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 8.8.4
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀⁴ dx / √(4 − x)
Guida verificata passo dopo passo1
Identify the integral to be solved: \(\int_0^4 \frac{dx}{\sqrt{4 - x}}\).
Recognize that the integrand involves a square root of a linear function, suggesting a substitution to simplify the integral.
Use the substitution \(u = 4 - x\), which implies $du = -dx$ or $dx = -du$. Also, change the limits of integration accordingly: when \(x=0\), \(u=4\); when \(x=4\), \(u=0\).
Rewrite the integral in terms of \(u\): \(\int_{u=4}^{u=0} \frac{-du}{\sqrt{u}} = \int_0^4 \frac{du}{\sqrt{u}}\) after reversing the limits to keep the integral positive.
Evaluate the integral \(\int_0^4 u^{-1/2} du\) by applying the power rule for integration: \(\int u^n du = \frac{u^{n+1}}{n+1} + C\) for \(n \neq -1\).
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Improper Integrals and Convergence
An integral is improper if the interval of integration is infinite or the integrand becomes unbounded within the interval. Understanding convergence means determining whether the integral approaches a finite value. In this problem, the integrand has a singularity at x = 4, so recognizing the integral as improper is essential.
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Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Substitution Method for Integration
The substitution method simplifies integrals by changing variables to transform the integrand into a more manageable form. For example, setting a new variable equal to the expression inside the square root can help rewrite the integral in terms of a simpler function, making it easier to evaluate.
Video consigliato:
07:33Euler's Method
Evaluating Definite Integrals with Square Root Functions
Integrals involving square roots often require algebraic manipulation or trigonometric substitution to evaluate. Recognizing the form √(a - x) allows the use of substitution or standard integral formulas to find antiderivatives and then apply the limits to compute the definite integral.
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Percorso guidato
05:43Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ sin(θ) sin(2θ) sin(3θ) dθ
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Domanda del libro di testo
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ p⁴ e^(-p) dp
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Domanda del libro di testo
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₋₁³ (4x² - 7) / (2x + 3) dx
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Domanda del libro di testo
Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx
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Domanda del libro di testo
Solve the initial value problems in Exercises 67–70 for x as a function of t.
(3t⁴ + 4t² + 1) (dx/dt) = 2√3, x(1) = -π√3/4
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