Textbook Question
Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.
b. To evaluate the integral ∫dx/√(x² − 100) analytically, it is best to use partial fractions.
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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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.R.76
76-81. Table of integrals Use a table of integrals to evaluate the following integrals.
76. ∫ x(2x + 3)⁵ dx
Verified step by step guidance1
Step 1: Recognize that the integral ∫ x(2x + 3)⁵ dx involves a product of a polynomial term x and a composite function (2x + 3) raised to a power. This suggests using a substitution method to simplify the integral.
Step 2: Let u = 2x + 3. Then, compute the derivative of u with respect to x: du/dx = 2, or equivalently, du = 2 dx. Rewrite dx in terms of du: dx = du/2.
Step 3: Substitute u and dx into the integral. Replace (2x + 3) with u and dx with du/2. Also, express x in terms of u using the substitution u = 2x + 3, which gives x = (u - 3)/2.
Step 4: After substitution, the integral becomes ∫ [(u - 3)/2] * u⁵ * (du/2). Simplify the constants and expand the expression to separate terms for easier integration.
Step 5: Use the table of integrals to evaluate the resulting integral term by term. For example, ∫ u⁶ du and ∫ u⁵ du can be directly integrated using standard formulas from the table of integrals. Combine the results and back-substitute u = 2x + 3 to express the solution in terms of x.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration
Integration is a fundamental concept in calculus that involves finding the antiderivative of a function. It is the process of calculating 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.
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Integration by Parts for Definite Integrals
Table of Integrals
A table of integrals is a reference tool that lists common integrals and their corresponding antiderivatives. It simplifies the process of integration by providing ready-made solutions for frequently encountered functions, allowing students to quickly find the integral of complex expressions without deriving them from first principles.
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08:09Tabular Integration by Parts
Polynomial Expansion
Polynomial expansion involves rewriting a polynomial expression in a more manageable form, often using the binomial theorem or distributive property. In the context of the given integral, expanding the expression x(2x + 3)⁵ will make it easier to integrate term by term, as it transforms the integral into a sum of simpler polynomial terms.
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07:00Taylor Polynomials
Related Practice
Textbook Question
114. {Use of Tech} Arc length of the natural logarithm Consider the curve y = ln(x).
a. Find the length of the curve from x = 1 to x = a and call it L(a).
(Hint: The change of variables u = sqrt(x^2 + 1) allows evaluation by partial fractions.)
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
3. ∫ (3x)/√(x + 4) dx
Textbook Question
Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.
d. ∫2 sin x cos x dx = −(1/2) cos 2x + C.
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Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
18. ∫ (from 0 to √2) (x + 1)/(3x² + 6) dx
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
9. ∫ (from 0 to π/4) cos⁵ 2x sin² 2x dx