Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₁^∞ (1 / x^(1/5)) dx
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.2.8
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫x e^(3x) dx
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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08:23Repeated Integration by Parts
Related Practice
Textbook Question
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))
Textbook Question
Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞² (2 dx) / (x² + 4)
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Textbook Question
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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Textbook Question
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