Domanda del libro di testo
Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77).
b. ∫ 7x e³ˣ 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 8.1.71b
71. Different Methods
Let I = ∫ (x²)/(x + 1) dx.
b. Evaluate I by first performing long division on the integrand.
Guida verificata passo dopo passo1
Perform long division on the integrand \( \frac{x^2}{x+1} \). Divide \( x^2 \) by \( x+1 \), which gives a quotient of \( x \) and a remainder of \( -x \). Rewrite the integrand as \( x - \frac{x}{x+1} \).
Split the integral \( I \) into two parts: \( I = \int x \, dx - \int \frac{x}{x+1} \, dx \).
Evaluate the first integral \( \int x \, dx \) using the power rule: \( \int x \, dx = \frac{x^2}{2} + C_1 \), where \( C_1 \) is a constant of integration.
For the second integral \( \int \frac{x}{x+1} \, dx \), perform substitution. Let \( u = x+1 \), so \( du = dx \) and \( x = u-1 \). Rewrite the integral as \( \int \frac{u-1}{u} \, du \), which simplifies to \( \int 1 \, du - \int \frac{1}{u} \, du \).
Evaluate the simplified integrals: \( \int 1 \, du = u \) and \( \int \frac{1}{u} \, du = \ln|u| \). Substitute back \( u = x+1 \) to get \( x+1 - \ln|x+1| + C_2 \), where \( C_2 \) is another constant of integration. Combine all results to express the final solution for \( I \).
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Long Division of Polynomials
Long division of polynomials is a method used to divide a polynomial by another polynomial of equal or lower degree. In the context of integration, this technique simplifies the integrand, allowing for easier evaluation of the integral. By dividing the numerator by the denominator, we can express the integrand as a sum of a polynomial and a proper fraction, which can be integrated separately.
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07:00
Taylor Polynomials
Integration Techniques
Integration techniques refer to various methods used to evaluate integrals, including substitution, integration by parts, and partial fraction decomposition. After simplifying the integrand through long division, the resulting polynomial can be integrated directly, while any remaining proper fraction may require additional techniques for evaluation. Understanding these methods is crucial for effectively solving integrals.
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Percorso guidato
06:18Integration by Parts for Definite Integrals
Definite vs. Indefinite Integrals
Indefinite integrals represent a family of functions whose derivatives yield the integrand, typically expressed with a constant of integration. In this problem, we are evaluating an indefinite integral, which means we will find the antiderivative of the simplified expression. Recognizing the difference between definite and indefinite integrals is essential for correctly interpreting the results of integration.
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Percorso guidato
05:43Definition of the Definite Integral
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
The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.
The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]
(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project ""The exponential Eiffel Tower"")
b. With a = 0 and c = 2, find the equations of the lines tangent to both curves at x = 0
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Domanda del libro di testo
Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.
b. Find the area of the region bounded by the graph of g and the x-axis on the interval [0,2].
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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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