Textbook Question
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ from 0 to 1 x√(1 - x) dx
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.8.32
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² dx / √|x − 1|
Verified step by step guidance1
First, recognize that the integral involves the expression \(\sqrt{|x - 1|}\), which has an absolute value inside the square root. This means the behavior of the integrand changes at \(x = 1\), so we should split the integral at this point.
Rewrite the integral as the sum of two integrals: \(\int_0^1 \frac{dx}{\sqrt{1 - x}} + \int_1^2 \frac{dx}{\sqrt{x - 1}}\). Notice how the absolute value affects the expression inside the square root on each interval.
For the first integral, \(\int_0^1 \frac{dx}{\sqrt{1 - x}}\), use the substitution \(u = 1 - x\), which implies \(du = -dx\). Adjust the limits accordingly and rewrite the integral in terms of \(u\).
For the second integral, \(\int_1^2 \frac{dx}{\sqrt{x - 1}}\), use the substitution \(v = x - 1\), which implies \(dv = dx\). Adjust the limits accordingly and rewrite the integral in terms of \(v\).
Evaluate both integrals using the power rule for integrals: \(\int u^{n} du = \frac{u^{n+1}}{n+1} + C\) for \(n \neq -1\). After evaluating, sum the results to get the value of the original integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals
Improper integrals involve integrands with infinite discontinuities or infinite limits of integration. In this problem, the integrand has a singularity at x = 1, where the denominator becomes zero, requiring careful evaluation of the integral as a limit approaching the point of discontinuity.
Recommended video:
11:11
Improper Integrals: Infinite Intervals
Absolute Value and Piecewise Functions
The absolute value inside the square root splits the integral into separate intervals where the expression inside changes sign. Understanding how to rewrite |x - 1| as (1 - x) or (x - 1) depending on the interval is essential to correctly set up and evaluate the integral.
Recommended video:
05:36Piecewise Functions
Integration Techniques for Power Functions
Integrating functions of the form 1/√(x - a) involves recognizing them as power functions with fractional exponents. Applying the power rule for integration, ∫x^n dx = x^(n+1)/(n+1), is necessary after rewriting the integrand appropriately to find the antiderivative.
Recommended video:
07:32Representing Functions as Power Series
Related Practice
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫₋₁¹ (√(1 + x²) sin x) dx
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Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ eˣ sec³(eˣ) dx
Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from -1 to 1 of (dθ / (θ² - 2θ))
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Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ x² √(1 - x²) dx
Textbook Question
Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.
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