Textbook Question
9–40. Integration by parts Evaluate the following integrals using integration by parts.
11. ∫ t · e⁶ᵗ dt
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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 8.7.46
41–48. Geometry problems Use a table of integrals to solve the following problems.
46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.
Verified step by step guidance1
Step 1: Recognize that the problem involves finding the area under the curve y = 1/√(x² - 2x + 2) from x = 0 to x = 3. This requires evaluating the definite integral of the function over the given interval.
Step 2: Simplify the quadratic expression x² - 2x + 2 in the denominator. Complete the square to rewrite it as (x - 1)² + 1. This step helps identify the structure of the integrand and makes it easier to match with a formula from the table of integrals.
Step 3: Refer to a table of integrals to find a formula that matches the form of the integrand. The expression 1/√((x - a)² + b²) corresponds to an arctangent integral formula: ∫ dx / √((x - a)² + b²) = (1/√b) * arctan((x - a)/√b) + C.
Step 4: Apply the formula to the given integral. Here, a = 1 and b² = 1, so √b = 1. Substitute these values into the formula to express the antiderivative of the function.
Step 5: Evaluate the definite integral by substituting the limits of integration (x = 0 and x = 3) into the antiderivative. Compute the difference between the values of the antiderivative at the upper and lower limits to find the area.
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4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
Definite integrals are used to calculate the area under a curve between two specified limits. In this problem, the area under the curve defined by the function y = 1/√(x² - 2x + 2) from x = 0 to x = 3 can be found by evaluating the definite integral of the function over that interval.
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05:43
Definition of the Definite Integral
Area Under a Curve
The area under a curve represents the integral of a function over a given interval. This area can be interpreted as the accumulation of values of the function, which in this case corresponds to the area between the curve and the x-axis from x = 0 to x = 3.
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Integral Tables
Integral tables are reference tools that provide a list of integrals and their solutions, which can simplify the process of finding areas or solving integrals. In this problem, using a table of integrals can help quickly identify the antiderivative of the function y = 1/√(x² - 2x + 2) needed to compute the definite integral.
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Related Practice
Textbook Question
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
8. ∫ sin 3x cos 2x dx
Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
39. ∫ x²/(100 - x²)^(3/2) dx
Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
26. ∫[√2 to √2] √(x² - 1)/x dx
Textbook Question
15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.
15. ∫(2 to 10) 2x² dx using n = 1, 2, and 4 subintervals
Textbook Question
67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
70. ∫ cos(x)cos(2x) dx