Textbook Question
23-64. Integration Evaluate the following integrals.
44. ∫₁² 2/[t³(t + 1)] dt
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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.3.25
9–61. Trigonometric integrals Evaluate the following integrals.
25. ∫ sin²x cos⁴x dx
Verified step by step guidance1
Step 1: Recognize that the integral involves powers of sine and cosine. For integrals of this type, it is often helpful to use trigonometric identities to simplify the expression. Specifically, use the Pythagorean identity: or , depending on the situation.
Step 2: Split the powers of sine and cosine into manageable parts. For example, rewrite as . This will allow us to work with smaller powers and apply substitution techniques.
Step 3: Use substitution to simplify the integral. Let , which implies . Rewrite the integral in terms of and . The integral becomes .
Step 4: Simplify the integral further. Distribute the terms and simplify: . Now, integrate each term separately using the power rule for integration: .
Step 5: After integrating, substitute back to return the solution to the original variable . Combine the terms and include the constant of integration .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Key identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas. These identities can simplify integrals involving trigonometric functions, making it easier to evaluate them.
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Verifying Trig Equations as Identities
Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric substitution. For integrals involving products of trigonometric functions, such as sin²x and cos⁴x, using substitution or recognizing patterns can significantly simplify the process.
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06:18Integration by Parts for Definite Integrals
Power Reduction Formulas
Power reduction formulas are used to express higher powers of sine and cosine in terms of first powers. For example, sin²x can be rewritten using the identity sin²x = (1 - cos(2x))/2. These formulas are particularly useful in integrals, as they transform the integrand into a more manageable form, facilitating easier integration.
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05:58Intro to Power Series
Related Practice
Textbook Question
"Electric field due to a line of charge A total charge of Q is distributed uniformly on a line segment of length 2L along the y-axis (see figure). The x-component of the electric field at a point (a, 0) is given by
Eₓ(a) = (kQa/2L) ∫-L L dy/(a² + y²)^(3/2),
where k is a physical constant and a > 0.
a. Confirm that Eₓ(a)=kQ / a √(a²+L²)
b. Letting ρ=Q / 2 L be the charge density on the line segment, show that if L → ∞, then Eₓ(a) = 2kρ / a.
Textbook Question
29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
30. ∫(0 to 6) (x³/16 - x) dx = 4
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Textbook Question
96. Challenge
Show that with the change of variables u = √tan x, the integral
∫ √tan x dx
can be converted to an integral amenable to partial fractions. Evaluate
∫[0 to π/4] √tan x dx.
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Textbook Question
76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.
79. ∫ [sec t / (1 + sin t)] dt
Textbook Question
76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.
82. ∫ [dx / (x√(1 + 2x))]