교과서 질문
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.
∫ dx / (x - √x)
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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
8장, 문제 8.2.38
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x⁵ e³ˣ dx
검증된 단계별 안내1
Identify the integral to solve: \(\int x^{5} e^{3x} \, dx\).
Recognize that this integral involves a product of a polynomial \(x^{5}\) and an exponential function \(e^{3x}\), which suggests using integration by parts.
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Choose \(u = x^{5}\) (since its derivative simplifies) and \(dv = e^{3x} \, dx\) (since it integrates easily).
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = 5x^{4} \, dx\), and integrate \(dv\) to get \(v = \frac{1}{3} e^{3x}\).
Apply the integration by parts formula: \(\int x^{5} e^{3x} \, dx = x^{5} \cdot \frac{1}{3} e^{3x} - \int \frac{1}{3} e^{3x} \cdot 5x^{4} \, dx\). This reduces the original integral to a similar integral with a lower power of \(x\), which you can solve by repeating integration by parts until the polynomial term is eliminated.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
5m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Integration by Parts
Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. This method is especially useful when integrating products like x⁵ and e³ˣ.
추천 영상:
06:18
Integration by Parts for Definite Integrals
Choosing u and dv
Selecting which part of the integrand to assign as u and which as dv is crucial for simplifying the integral. Typically, u is chosen as a polynomial (like x⁵) because its derivative simplifies, and dv as an exponential or trigonometric function, which is easy to integrate.
추천 영상:
07:51Choosing a Convergence Test
Repeated Application of Integration by Parts
When the polynomial degree is high, integration by parts may need to be applied multiple times, reducing the power of x step-by-step. This iterative process continues until the polynomial term is eliminated, allowing the integral to be evaluated fully.
추천 영상:
08:23Repeated Integration by Parts
관련 실천
교과서 질문
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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교과서 질문
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ csc³(√θ) / √θ dθ
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교과서 질문
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ ln(x + x²) dx
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교과서 질문
Use any method to evaluate the integrals in Exercises 65–70.
∫ sin³(x) / cos⁴(x) dx
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교과서 질문
Solve the initial value problems in Exercises 53–56 for y as a function of x.
√(x² - 9) (dy/dx) = 1, where x > 3, y(5) = ln 3
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