Textbook Question
23-64. Integration Evaluate the following integrals.
44. ∫₁² 2/[t³(t + 1)] dt
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 8.5.66
66-68. Areas of regions (Use of Tech) Find the area of the following regions.
66. The region bounded by the curve y = (x - x²)/[(x + 1)(x² + 1)] and the x-axis from x = 0 to x = 1
Verified step by step guidance1
Identify the function to integrate: \( y = \frac{(x - x^{2})}{(x + 1)(x^{2} + 1)} \). We want to find the area between this curve and the x-axis from \( x = 0 \) to \( x = 1 \).
Set up the definite integral for the area: \[ \text{Area} = \int_{0}^{1} \left| \frac{(x - x^{2})}{(x + 1)(x^{2} + 1)} \right| \, dx \]. Since area is always positive, we take the absolute value of the function inside the integral.
Determine where the function \( y \) changes sign on the interval \( [0,1] \) by solving \( y = 0 \). This means solving \( x - x^{2} = 0 \) because the denominator is always positive for \( x \geq 0 \). The roots are \( x = 0 \) and \( x = 1 \), so check the sign of \( y \) between these points to see if the function is positive or negative.
If the function does not change sign between 0 and 1, you can remove the absolute value and integrate \( y \) directly. Otherwise, split the integral at the points where \( y = 0 \) and integrate the absolute value accordingly.
Use technology (such as a graphing calculator or computer algebra system) to evaluate the definite integral(s) set up in the previous step to find the exact area.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral for Area Calculation
The definite integral of a function over an interval [a, b] represents the net area between the curve and the x-axis. When the function is positive, the integral gives the area directly; if negative, it subtracts area. To find the total area bounded by the curve and the x-axis, one may need to consider absolute values or split the integral at points where the function crosses the axis.
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Definition of the Definite Integral
Analyzing the Sign of the Function
Before integrating, it is important to determine where the function is positive or negative within the interval. This helps in correctly calculating the area, as areas below the x-axis contribute negatively to the integral. Identifying zeros of the function within the interval allows splitting the integral accordingly.
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Use of Technology in Integration
Complex rational functions like y = (x - x²)/[(x + 1)(x² + 1)] may be difficult to integrate by hand. Technology tools such as graphing calculators or computer algebra systems can assist in evaluating definite integrals accurately and efficiently, as well as in visualizing the function to understand its behavior over the interval.
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Related Practice
Textbook Question
9–61. Trigonometric integrals Evaluate the following integrals.
34. ∫ tan⁹x sec⁴x dx
Textbook Question
7–64. Integration review Evaluate the following integrals.
24. ∫ from 0 to θ of (x⁵⸍² - x¹⸍²) / x³⸍² dx
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Textbook Question
29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules
Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.
30. ∫(0 to 6) (x³/16 - x) dx = 4
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Textbook Question
76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.
82. ∫ [dx / (x√(1 + 2x))]
Textbook Question
64. Using a computer algebra system, it was determined that
∫x(x+1)^8 dx = (x^10)/10 + (8x^9)/9 + (7x^8)/2 + 8x^7 + (35x^6)/3 + (56x^5)/5 + 7x^4 + (8x^3)/3 + x^2/2 + C.
Use integration by substitution to evaluate ∫x(x+1)^8 dx.