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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 8 - Techniques of IntegrationProblem 8.PE.29b
Chapter 8, Problem 8.PE.29b

Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [y / √(16 − y²)] dy

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Identify the integral to solve: \(\int \frac{y}{\sqrt{16 - y^{2}}} \, dy\).
Recognize that the integrand contains a square root of the form \(\sqrt{a^{2} - y^{2}}\), which suggests using the trigonometric substitution \(y = a \sin(\theta)\), where \(a = 4\) in this case.
Make the substitution \(y = 4 \sin(\theta)\), then compute \(dy = 4 \cos(\theta) \, d\theta\). Also, rewrite the square root: \(\sqrt{16 - y^{2}} = \sqrt{16 - 16 \sin^{2}(\theta)} = 4 \cos(\theta)\).
Substitute all parts into the integral: replace \(y\), \(dy\), and the square root to express the integral entirely in terms of \(\theta\). The integral becomes \(\int \frac{4 \sin(\theta)}{4 \cos(\theta)} \times 4 \cos(\theta) \, d\theta\).
Simplify the integral expression and then integrate with respect to \(\theta\). After integration, use the inverse substitution \(\theta = \arcsin\left(\frac{y}{4}\right)\) to rewrite the answer in terms of \(y\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots of quadratic expressions by substituting a trigonometric function for the variable. For expressions like √(a² − x²), substituting x = a sin(θ) transforms the integral into a trigonometric form that is easier to evaluate.
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Introduction to Trigonometric Functions

Integration of Trigonometric Functions

After substitution, the integral often involves trigonometric functions such as sine and cosine. Understanding how to integrate these functions, including using identities and basic integral formulas, is essential to solve the integral and then revert back to the original variable.
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Introduction to Trigonometric Functions

Back-Substitution

Once the integral is evaluated in terms of the trigonometric variable, back-substitution is used to rewrite the answer in terms of the original variable. This involves using the inverse trigonometric functions or right triangle relationships derived from the substitution.
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Substitution With an Extra Variable
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