Textbook Question
Practice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77).
b. ∫ 7x e³ˣ dx
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 8.1.71b
71. Different Methods
Let I = ∫ (x²)/(x + 1) dx.
b. Evaluate I by first performing long division on the integrand.
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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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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Related Practice
Textbook Question
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).
Textbook Question
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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Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
69. Let f(x) = sin(eˣ).
a. Find a Trapezoid Rule approximation to ∫[0 to 1] sin(eˣ) dx using n = 40 subintervals.
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Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
a. Find a Midpoint Rule approximation to ∫[1 to 6] √(x³ + 1) dx using n = 50 subintervals.
Textbook Question
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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