Textbook Question
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)
1
views
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.6.10
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x - x²) / x dx
Verified step by step guidance1
Start by rewriting the integrand to a more manageable form. Notice that the expression under the square root is \(x - x^2\), which can be factored as \(x(1 - x)\). So the integral becomes \(\int \frac{\sqrt{x(1 - x)}}{x} \, dx\).
Simplify the integrand by separating the square root and dividing by \(x\): \(\frac{\sqrt{x(1 - x)}}{x} = \frac{\sqrt{x} \sqrt{1 - x}}{x} = \frac{\sqrt{1 - x}}{\sqrt{x}}\).
Rewrite the integral as \(\int \frac{\sqrt{1 - x}}{\sqrt{x}} \, dx = \int \frac{(1 - x)^{1/2}}{x^{1/2}} \, dx = \int x^{-1/2} (1 - x)^{1/2} \, dx\).
To evaluate this integral, consider a substitution that simplifies the expression. A common substitution for integrals involving \(\sqrt{1 - x}\) is \(x = \sin^2 \theta\), because \(1 - \sin^2 \theta = \cos^2 \theta\). This substitution will transform the integral into a trigonometric integral.
After substituting \(x = \sin^2 \theta\), express \(dx\) in terms of \(d\theta\), rewrite the integral entirely in terms of \(\theta\), and then use the table of integrals to evaluate the resulting trigonometric integral.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration Using Substitution
Substitution is a method to simplify integrals by changing variables, making the integral easier to evaluate. It often involves identifying a part of the integrand whose derivative also appears, allowing a direct replacement. This technique is essential when dealing with composite functions or expressions under roots.
Recommended video:
04:27
Substitution With an Extra Variable
Using Integral Tables
Integral tables provide formulas for common integrals, saving time and effort in manual integration. Recognizing the form of the integrand and matching it to a formula in the table is crucial. This skill helps in quickly evaluating integrals that might be complicated to solve from first principles.
Recommended video:
08:01Integration Using Partial Fractions
Simplifying the Integrand
Before integrating, simplifying the integrand by algebraic manipulation or rewriting expressions can make the integral more manageable. For example, rewriting √(x - x²) as √(x(1 - x)) can suggest substitutions or reveal standard integral forms. Simplification is a key step in preparing the integral for substitution or table lookup.
Recommended video:
05:22Completing the Square to Rewrite the Integrand
Related Practice
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x (7x + 5)^(3/2) 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 ∞ of ((1 / (e^x - 2^x)) dx)
1
views
Textbook Question
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ √(x² + 2x + 2) / (x² + 2x + 1) dx
Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x dx) / (25 + 4x²)
1
views
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x⁵ e³ˣ dx