Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
60. ∫ x² coshx 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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.R.3
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
Verified step by step guidance1
Step 1: Recognize that the integral ∫ (3x)/√(x + 4) dx involves a combination of algebraic and radical expressions. To simplify, consider substitution techniques to reduce the complexity of the integrand.
Step 2: Let u = x + 4. Then, compute the derivative of u with respect to x: du/dx = 1, which implies that du = dx. Also, note that x = u - 4.
Step 3: Substitute u and du into the integral. Replace x with (u - 4) and √(x + 4) with √u. The integral becomes ∫ (3(u - 4))/√u du.
Step 4: Split the integral into two simpler parts: ∫ (3u/√u) du - ∫ (12/√u) du. Simplify each term using the property √u = u^(1/2). For the first term, 3u/√u simplifies to 3u^(1/2). For the second term, 12/√u simplifies to 12u^(-1/2).
Step 5: Integrate each term separately. Use the power rule for integration: ∫ u^n du = (u^(n+1))/(n+1) + C, where n ≠ -1. For the first term, integrate 3u^(1/2). For the second term, integrate -12u^(-1/2). Combine the results and substitute back u = x + 4 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 Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fractions. Understanding these methods is crucial for evaluating more complex integrals, as they allow for simplification and manipulation of the integrand to make integration feasible.
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Integration by Parts for Definite Integrals
Substitution Method
The substitution method is a technique where a new variable is introduced to simplify the integral. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative.
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07:33Euler's Method
Definite vs. Indefinite Integrals
Definite integrals calculate the area under a curve between two specific limits, while indefinite integrals represent a family of functions and include a constant of integration. Understanding the difference is essential for correctly applying integration techniques, as the approach may vary depending on whether the integral is definite or indefinite.
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05:43Definition of the Definite Integral
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
110. Comparing distances Suppose two cars started at the same time and place (t = 0 and s = 0). The velocity of car A (in mi/hr) is given by
u(t) = 40 / (t + 1) and the velocity of car B (in mi/hr) is given by v(t) = 40 * e^(-t/2).
b. After t = 3 hr, which car has traveled farther?
Textbook Question
123. Region between curves Find the area of the region bounded by the graphs of y = tan(x) and y = sec(x) on the interval [0, π/4].
Textbook Question
76-81. Table of integrals Use a table of integrals to evaluate the following integrals.
76. ∫ x(2x + 3)⁵ dx
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