교과서 질문
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (4 - x²)^(3/2) from 0 to 1
23
views
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
당신이 사용하는 게 아니라요?교과서 변경다른 교과서를 선택하세요
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
8장, 문제 8.2.32
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (cos(√x))/(√x) dx
검증된 단계별 안내1
Start by identifying a suitable substitution to simplify the integral. Notice that the integrand contains both \( \cos(\sqrt{x}) \) and \( \frac{1}{\sqrt{x}} \), which suggests substituting \( u = \sqrt{x} \).
Express \( u = \sqrt{x} \) in terms of \( x \), so \( u = x^{1/2} \). Then, differentiate both sides with respect to \( x \) to find \( du \): \( du = \frac{1}{2} x^{-1/2} dx = \frac{1}{2\sqrt{x}} dx \).
Solve for \( dx \) in terms of \( du \) and \( x \): \( dx = 2 \sqrt{x} du \). Substitute \( dx \) and \( \sqrt{x} = u \) back into the integral to rewrite it entirely in terms of \( u \).
After substitution, the integral becomes \( \int \frac{\cos(u)}{u} \cdot 2u \, du \). Simplify the expression inside the integral by canceling terms where possible.
Integrate the simplified integral with respect to \( u \), then substitute back \( u = \sqrt{x} \) to express the answer in terms of \( x \).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Substitution Method
The substitution method simplifies integrals by changing variables to transform the integral into a more familiar form. For example, setting a new variable equal to a function inside the integral can make the integral easier to evaluate.
추천 영상:
07:33
Euler's Method
Integration of Trigonometric Functions
Understanding how to integrate trigonometric functions like cosine is essential. The integral of cos(u) with respect to u is sin(u) plus a constant, which helps in solving integrals involving trigonometric expressions.
추천 영상:
가이드 코스
6:04Introduction to Trigonometric Functions
Handling Radicals in Integrals
Integrals involving radicals, such as square roots, often require rewriting the expression in terms of powers or using substitution to simplify the integral. Recognizing how to manipulate radicals is key to evaluating such integrals.
추천 영상:
06:13Limits of Rational Functions with Radicals
관련 실천
교과서 질문
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = x²/4, 0 ≤ x ≤ 2
28
views
교과서 질문
Finding volume
The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.
35
views
교과서 질문
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ 2x e^(−x²) dx
37
views
교과서 질문
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x + 3) / (2x³ - 8x) dx
19
views
교과서 질문
For each x > 0, let G(x) = ∫(from 0 to x) e^(-xt) dt. Prove that xG(x) = 1 for each x > 0.
17
views