Domanda del libro di testo
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
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.R.60
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
60. ∫ x² coshx dx
Guida verificata passo dopo passo1
Step 1: Recognize that the integral ∫ x² cosh(x) dx involves a product of a polynomial (x²) and a hyperbolic function (cosh(x)). This suggests that the method of integration by parts is appropriate. Recall the formula for integration by parts: ∫ u dv = uv - ∫ v du.
Step 2: Choose u and dv wisely. Let u = x² (the polynomial term, which simplifies upon differentiation) and dv = cosh(x) dx (the hyperbolic function, which is straightforward to integrate). Compute du = 2x dx and v = sinh(x), since the integral of cosh(x) is sinh(x).
Step 3: Apply the integration by parts formula: ∫ x² cosh(x) dx = u * v - ∫ v * du. Substitute u = x², v = sinh(x), and du = 2x dx into the formula to get: x² sinh(x) - ∫ 2x sinh(x) dx.
Step 4: Notice that the remaining integral ∫ 2x sinh(x) dx still involves a product of a polynomial and a hyperbolic function. Apply integration by parts again. Let u = 2x and dv = sinh(x) dx. Compute du = 2 dx and v = -cosh(x), since the integral of sinh(x) is -cosh(x).
Step 5: Substitute into the integration by parts formula for the second integral: ∫ 2x sinh(x) dx = u * v - ∫ v * du. This becomes 2x(-cosh(x)) - ∫ -cosh(x) * 2 dx. Simplify the expression and combine terms to complete the solution.
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Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric identities. Understanding these methods is crucial for solving integrals that cannot be evaluated using basic antiderivatives.
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Percorso guidato
06:18
Integration by Parts for Definite Integrals
Hyperbolic Functions
Hyperbolic functions, such as cosh(x), are analogs of trigonometric functions but are based on hyperbolas rather than circles. They are defined using exponential functions, with cosh(x) = (e^x + e^(-x))/2. Recognizing these functions and their properties is essential for evaluating integrals involving them.
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5:50Asymptotes of Hyperbolas
Integration by Parts
Integration by parts is a technique derived from the product rule of differentiation. It is used to integrate products of functions and is expressed as ∫u dv = uv - ∫v du. This method is particularly useful when dealing with integrals that involve polynomial and hyperbolic functions, as in the given problem.
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06:18Integration by Parts for Definite Integrals
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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?
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Domanda del libro di testo
106. Arc length Find the length of the curve y = (x / 2) * sqrt(3 - x^2) + (3 / 2) * sin^(-1)(x / sqrt(3)) from x = 0 to x = 1.
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Domanda del libro di testo
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
74. ∫ dx/√(√(1 + √x))
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Domanda del libro di testo
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].
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Domanda del libro di testo
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
68. ∫ (from -1 to 1) dx/(x² + 2x + 5)
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