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 / √(2θ - θ²))
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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 8.2.8
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫x e^(3x) dx
Guida verificata passo dopo passo1
Identify the parts of the integral for integration by parts. Let \( u = x \) (a polynomial function) and \( dv = e^{3x} dx \) (an exponential function).
Compute the derivatives and integrals needed: find \( du = dx \) by differentiating \( u \), and find \( v = \int e^{3x} dx \). To integrate \( e^{3x} \), recall that \( \int e^{ax} dx = \frac{1}{a} e^{ax} + C \).
Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the expressions for \( u, v, du \) into this formula.
Set up the resulting integral after substitution, which should be simpler to evaluate than the original integral. This will involve integrating an exponential function.
Evaluate the remaining integral and combine all terms. Don't forget to add the constant of integration \( + C \) at the end.
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Integration by Parts
Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely simplifies the integration process.
Video consigliato:
Percorso guidato
06:18
Integration by Parts for Definite Integrals
Exponential Functions
Exponential functions have the form e^(kx), where k is a constant. Their derivatives and integrals are proportional to the original function, making them straightforward to handle in integration, especially when combined with polynomial terms.
Video consigliato:
6:13Exponential Functions
Choosing u and dv in Integration by Parts
Selecting u and dv correctly is crucial for simplifying the integral. Typically, u is chosen as a polynomial or algebraic function that simplifies upon differentiation, while dv is chosen as the remaining part that is easy to integrate, such as an exponential function.
Video consigliato:
Percorso guidato
08:23Repeated Integration by Parts
Pratica correlata
Domanda del libro di testo
Evaluate the integrals in Exercises 33–52.
∫ from -π/4 to π/4 of 6 tan⁴(x) dx
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Domanda del libro di testo
Evaluate the integrals in Exercises 39–54.
∫ 1 / ((x¹/³ - 1)√x) 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.
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Domanda del libro di testo
What is the largest value that
∫ from a to b x√(2x - x²) dx
can have for any a and b? Give reasons for your answer.
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Domanda del libro di testo
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ x √(x² - 4) dx
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