Domanda del libro di testo
71. Different Methods
Let I = ∫ (x²)/(x + 1) dx.
b. Evaluate I by first performing long division on the integrand.
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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 8.2.78b
Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77).
b. ∫ 7x e³ˣ dx
Guida verificata passo dopo passo1
Identify the parts of the integrand for tabular integration: choose the algebraic function to differentiate and the exponential function to integrate. Here, let \( u = 7x \) (to differentiate) and \( dv = e^{3x} dx \) (to integrate).
Create a table with two columns: one for successive derivatives of \( u = 7x \), and one for successive integrals of \( dv = e^{3x} dx \). Differentiate \( u \) until it reaches zero, and integrate \( dv \) the same number of times.
Multiply diagonally across the table, alternating signs starting with positive. This means the first product is positive, the second is negative, and so on. Write these products as terms in the integral's expression.
Sum all the diagonal products obtained from the table to form the expression for the integral, and add the constant of integration \( + C \) at the end.
Review the result to ensure all derivatives and integrals are correctly computed, and the signs alternate properly. This completes the tabular integration process for \( \int 7x e^{3x} dx \).
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Integration by Parts
Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely simplifies the integration process.
Video consigliato:
Percorso guidato
06:18
Integration by Parts for Definite Integrals
Tabular Integration Method
Tabular integration is a streamlined approach to repeated integration by parts, especially useful when one function differentiates to zero after several steps. It organizes derivatives and integrals in a table, allowing quick computation of the integral without writing each step explicitly.
Video consigliato:
Percorso guidato
09:15Tabular Integration by Parts Example 6
Exponential and Polynomial Functions in Integration
When integrating products of polynomial and exponential functions, the polynomial is typically chosen as u to differentiate down to zero, while the exponential is integrated repeatedly. This strategy simplifies the integral and is well-suited for tabular integration.
Video consigliato:
05:11Integrals of General Exponential Functions
Pratica correlata
Domanda del libro di testo
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
66. Let f(x) = cos(x²).
b. Calculate f''(x).
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Domanda del libro di testo
{Use of Tech} Powers of sine and cosine It can be shown that
∫ from 0 to π/2 of sinⁿx dx = ∫ from 0 to π/2 of cosⁿx dx =
{
[1·3·5···(n-1)]/[2·4·6···n] · π/2 if n ≥ 2 is even
[2·4·6···(n-1)]/[3·5···n] if n ≥ 3 is odd
}
b. Evaluate the integrals with n = 10 and confirm the result.
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Domanda del libro di testo
93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:
vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).
b. Which car travels farthest on the interval 0 ≤ t ≤ 5?
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Domanda del libro di testo
Area and volume Consider the function f(x) = (9 + x²)^(-1/2) and the region R on the interval [0, 4] (see figure).
b. Find the volume of the solid generated when R is revolved about the x-axis.
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Domanda del libro di testo
68. Different methods
b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.
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