Textbook Question
Differentiating Integrals
In Exercises 75–78, find dy/dx.
________
y = ∫₂ˣ √ 2 + cos³t dt
Ch. 5 - Integrals
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 5, Problem 5.PE.37
Evaluating Indefinite Integrals
Evaluate the integrals in Exercises 37–46.
∫ 2(cos x)⁻¹/² sin x dx
Verified step by step guidance1
Recognize that the integral is of the form \(\int 2 (\cos x)^{-\frac{1}{2}} \sin x \, dx\), which suggests a substitution involving \(\cos x\) because its derivative is related to \(\sin x\).
Let \(u = \cos x\). Then, compute the differential \(du = -\sin x \, dx\), which implies \(-du = \sin x \, dx\).
Rewrite the integral in terms of \(u\): substitute \(\cos x\) with \(u\) and \(\sin x \, dx\) with \(-du\), so the integral becomes \(\int 2 u^{-\frac{1}{2}} (-du)\).
Simplify the integral to \(-2 \int u^{-\frac{1}{2}} \, du\), which is a basic power integral.
Use the power rule for integration: \(\int u^{n} \, du = \frac{u^{n+1}}{n+1} + C\) for \(n \neq -1\), and apply it to \(u^{-\frac{1}{2}}\) to find the antiderivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integrals
An indefinite integral represents the family of all antiderivatives of a function and is expressed without limits of integration. It includes a constant of integration (C) because differentiation of a constant is zero. Understanding indefinite integrals is essential for reversing differentiation and finding original functions.
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Introduction to Indefinite Integrals
Integration by Substitution
Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable (u), rewriting the integral in terms of u, and then integrating. This technique is especially useful when the integral contains a function and its derivative.
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04:27Substitution With an Extra Variable
Trigonometric Functions and Their Integrals
Trigonometric functions like sine and cosine have specific integral formulas and properties. Recognizing how to manipulate powers and compositions of trig functions is crucial. For example, integrating expressions involving powers of cosine or sine often requires substitution or trigonometric identities.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.
a. ∫²₋₂ ƒ(x) dx
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Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
x = 2y², x = 0, y = 3
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Find the extreme values of ƒ(x) = x³ - 3x², and find the area of the region enclosed by the graph of ƒ and the x-axis.
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Textbook Question
Find dy/dx if y = ∫ₓ¹ √(1 + t²)dt.
Explain the main steps in your calculation.
Textbook Question
If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.
e. ∫⁵₋₂ ( ƒ(x) + g(x) ) dx
5
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