교과서 질문
7–84. Evaluate the following integrals.
11. ∫ from 0 to π/4 (sec x – cos x)² dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
8장, 문제 8.1.10
7–64. Integration review Evaluate the following integrals.
10. ∫ e^(3 - 4x) dx
검증된 단계별 안내1
Step 1: Recognize that the integral involves an exponential function, e^(3 - 4x). To simplify, identify the inner function (3 - 4x) and its derivative (-4). This suggests using substitution.
Step 2: Perform substitution. Let u = 3 - 4x, which implies that du/dx = -4 or equivalently, dx = -du/4.
Step 3: Rewrite the integral in terms of u. Substituting u and dx, the integral becomes ∫ e^u * (-du/4). Factor out the constant -1/4 to simplify: (-1/4) ∫ e^u du.
Step 4: Integrate e^u with respect to u. The integral of e^u is simply e^u, so the result becomes (-1/4) * e^u + C, where C is the constant of integration.
Step 5: Substitute back u = 3 - 4x to return to the original variable. The final expression is (-1/4) * e^(3 - 4x) + C.
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주요 개념
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Integration
Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is the process of determining the area under a curve represented by a function over a specified interval. Understanding integration is crucial for evaluating integrals, as it allows us to reverse the process of differentiation.
추천 영상:
06:18
Integration by Parts for Definite Integrals
Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where 'e' is the base of natural logarithms, approximately equal to 2.71828. These functions are characterized by their rapid growth or decay and are commonly encountered in calculus, especially in integration problems involving e raised to a power. Recognizing the structure of exponential functions is essential for applying integration techniques.
추천 영상:
6:13Exponential Functions
Substitution Method
The substitution method is a technique used in integration to simplify the process of finding an integral. It involves substituting a part of the integrand with a new variable to make the integral easier to evaluate. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative, allowing for a more straightforward integration process.
추천 영상:
07:33Euler's Method
관련 실천
교과서 질문
42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.
44. The region bounded by f(x) = sin(x) and the x-axis on [0, π] is revolved about the y-axis.
57
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교과서 질문
69-72. Volumes of solids Find the volume of the following solids.
70. The region bounded by y = 1/[x²(x² + 2)²], y = 0, x = 1, and x = 2 is revolved about the y-axis.
48
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교과서 질문
50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:
51. ∫ xⁿ cos(ax) dx = (xⁿ sin(ax))/a - (n/a) ∫ xⁿ⁻¹ sin(ax) dx, for a ≠ 0
93
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교과서 질문
9–61. Trigonometric integrals Evaluate the following integrals.
28. ∫ 6 sec⁴x dx
84
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교과서 질문
9–61. Trigonometric integrals Evaluate the following integrals.
22. ∫[π/4 to π/2] sin²(2x) cos³(2x) dx
78
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