Domanda del libro di testo
Evaluate the integrals in Exercises 53–58.
∫ from -π/2 to π/2 of cos(x) cos(7x) dx
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.2.38
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x⁵ e³ˣ dx
Guida verificata passo dopo passo1
Identify the integral to solve: \(\int x^{5} e^{3x} \, dx\).
Recognize that this integral involves a product of a polynomial \(x^{5}\) and an exponential function \(e^{3x}\), which suggests using integration by parts.
Recall the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Choose \(u = x^{5}\) (since its derivative simplifies) and \(dv = e^{3x} \, dx\) (since it integrates easily).
Compute \(du\) and \(v\): differentiate \(u\) to get \(du = 5x^{4} \, dx\), and integrate \(dv\) to get \(v = \frac{1}{3} e^{3x}\).
Apply the integration by parts formula: \(\int x^{5} e^{3x} \, dx = x^{5} \cdot \frac{1}{3} e^{3x} - \int \frac{1}{3} e^{3x} \cdot 5x^{4} \, dx\). This reduces the original integral to a similar integral with a lower power of \(x\), which you can solve by repeating integration by parts until the polynomial term is eliminated.
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Integration by Parts
Integration by parts is a technique based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. This method is especially useful when integrating products like x⁵ and e³ˣ.
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Percorso guidato
06:18
Integration by Parts for Definite Integrals
Choosing u and dv
Selecting which part of the integrand to assign as u and which as dv is crucial for simplifying the integral. Typically, u is chosen as a polynomial (like x⁵) because its derivative simplifies, and dv as an exponential or trigonometric function, which is easy to integrate.
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Percorso guidato
07:51Choosing a Convergence Test
Repeated Application of Integration by Parts
When the polynomial degree is high, integration by parts may need to be applied multiple times, reducing the power of x step-by-step. This iterative process continues until the polynomial term is eliminated, allowing the integral to be evaluated fully.
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Percorso guidato
08:23Repeated Integration by Parts
Pratica correlata
Domanda del libro di testo
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
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Domanda del libro di testo
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
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Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
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Domanda del libro di testo
Evaluate the integrals in Exercises 33–52.
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Domanda del libro di testo
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
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