Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ sec(x) tan²(x) dx
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 8.1.26
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.
∫ (6 dy / √y(1 + y))
Verified step by step guidance1
Rewrite the integral to make it clearer: \(\int \frac{6}{\sqrt{y}(1 + y)} \, dy\).
Express \(\sqrt{y}\) as \(y^{1/2}\) and rewrite the integral as \(\int \frac{6}{y^{1/2}(1 + y)} \, dy\).
Consider the substitution \(y = t^2\), so that \(dy = 2t \, dt\) and \(\sqrt{y} = t\).
Rewrite the integral in terms of \(t\): replace \(y\) and \(dy\) accordingly, then simplify the expression.
After substitution, simplify the integral and look for a method to integrate, such as partial fractions or a direct algebraic simplification.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Substitution
Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a part of the integrand, the integral can become easier to evaluate. This method is especially useful when the integrand contains composite functions or expressions under radicals.
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Substitution With an Extra Variable
Algebraic Manipulation of Integrands
Algebraic manipulation includes rewriting the integrand to a more convenient form, such as factoring, expanding, or simplifying expressions. This step can reveal standard integral forms or make substitution more straightforward, facilitating the integration process.
Recommended video:
05:22Completing the Square to Rewrite the Integrand
Properties of Radicals and Exponents
Understanding how to handle radicals and fractional exponents is crucial when integrating functions involving roots. Converting radicals to fractional powers allows the use of power rule integration, and recognizing how to simplify expressions under the root helps in choosing the right substitution.
Recommended video:
06:21Properties of Functions
Related Practice
Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ dy / (y√(1 + (ln y)²)) from 1 to e
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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 -∞ to ∞ of ((dx) / (e^x + e^(-x)))
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Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ √(1 - 9t²) dt
Textbook Question
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 2 sin^2(t) sec^4(t) dt
Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ sec³(x) tan³(x) dx