Domanda del libro di testo
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (3t² + t + 4) / (t³ + t) dt from 1 to √3
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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 8.3.14
Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(x) dx
Guida verificata passo dopo passo1
Recognize that the integral involves the square of the sine function: \(\int_0^{\frac{\pi}{2}} \sin^2(x) \, dx\).
Use the power-reduction identity to rewrite \(\sin^2(x)\) in a more integrable form: \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\).
Substitute this identity into the integral to get \(\int_0^{\frac{\pi}{2}} \frac{1 - \cos(2x)}{2} \, dx\).
Split the integral into two separate integrals: \(\frac{1}{2} \int_0^{\frac{\pi}{2}} 1 \, dx - \frac{1}{2} \int_0^{\frac{\pi}{2}} \cos(2x) \, dx\).
Evaluate each integral separately using basic integration rules, then combine the results to express the value of the original integral.
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Definite Integral
A definite integral calculates the net area under a curve between two specified limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating a definite integral involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.
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Percorso guidato
05:43
Definition of the Definite Integral
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. For integrals like ∫ sin²(x) dx, using identities such as the power-reduction formula sin²(x) = (1 - cos(2x))/2 simplifies the integral into a more manageable form.
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7:17Verifying Trig Equations as Identities
Fundamental Theorem of Calculus
This theorem links differentiation and integration, stating that if F is an antiderivative of f on [a, b], then ∫_a^b f(x) dx = F(b) - F(a). It allows evaluation of definite integrals by finding antiderivatives and computing their difference at the bounds.
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06:11Fundamental Theorem of Calculus Part 1
Pratica correlata
Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (xe^x) / (x + 1)² dx
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Domanda del libro di testo
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to 2 of (dx / (1 - x))
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Domanda del libro di testo
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Use numerical integration to estimate the value of
π = 4 ∫ (from 0 to 1) [ 1 / (1 + x²) ] dx.
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Domanda del libro di testo
Evaluate the integrals in Exercises 53–58.
∫ sin(2x) cos(3x) dx
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Domanda del libro di testo
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ dx / (x² + 2x)
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