Domanda del libro di testo
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(1 - (ln x)²) / (x ln x) dx
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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 8.1.14
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.
∫ (csc t sin 3t dt)
Guida verificata passo dopo passo1
Start by rewriting the integral \( \int \csc t \sin 3t \, dt \) to understand the integrand better. Recall that \( \csc t = \frac{1}{\sin t} \), so the integrand becomes \( \frac{\sin 3t}{\sin t} \).
Use the trigonometric identity for \( \sin 3t \), which is \( \sin 3t = 3 \sin t - 4 \sin^3 t \). Substitute this into the integrand to get \( \frac{3 \sin t - 4 \sin^3 t}{\sin t} \).
Simplify the fraction by dividing each term in the numerator by \( \sin t \), resulting in \( 3 - 4 \sin^2 t \). Now the integral becomes \( \int (3 - 4 \sin^2 t) \, dt \).
Recall the Pythagorean identity \( \sin^2 t = \frac{1 - \cos 2t}{2} \). Substitute this into the integral to express everything in terms of cosine: \( \int \left(3 - 4 \cdot \frac{1 - \cos 2t}{2} \right) dt \).
Simplify the integrand and then integrate term-by-term. This will involve integrating constants and cosine functions, which are straightforward to handle.
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They simplify expressions and integrals by rewriting complex trigonometric terms into more manageable forms, such as expressing sin(3t) using angle addition formulas.
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7:17
Verifying Trig Equations as Identities
Integration Techniques
Integration techniques include methods like substitution, integration by parts, and algebraic manipulation. Choosing the right technique helps transform the integral into a simpler form that can be directly integrated, especially when dealing with products of trigonometric functions.
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Percorso guidato
06:18Integration by Parts for Definite Integrals
Properties of Cosecant and Sine Functions
Understanding the properties of cosecant (csc t = 1/sin t) and sine functions is essential to simplify the integrand. Recognizing how these functions interact allows for algebraic simplification, such as canceling terms or rewriting the integrand to facilitate integration.
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Percorso guidato
06:21Properties of Functions
Pratica correlata
Domanda del libro di testo
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
85. Find the volume of the solid generated by revolving the region about the y-axis.
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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 4 to ∞ of (dx / (√x - 1))
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Domanda del libro di testo
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (8x² + 8x + 2) / (4x² + 1)² dx
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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.
∫ (dθ / cos θ - 1)
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Domanda del libro di testo
Evaluate the integrals in Exercises 1–14.
∫ (2 dx) / √(1 - 4x²) from 0 to 1/(2√2)
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