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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 8.4.18
Chapter 8, Problem 8.4.18

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / √(1 - x²)

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1
Recognize that the integral \( \int \frac{dx}{\sqrt{1 - x^2}} \) resembles the derivative of an inverse trigonometric function, specifically \( \arcsin x \) or \( \arccos x \).
Recall the derivative formula: \( \frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}} \). This suggests that the integral is related to \( \arcsin x \).
Set up the integral using the substitution method if needed, but in this case, direct recognition is sufficient since the integrand matches the derivative of \( \arcsin x \).
Write the integral as \( \int \frac{dx}{\sqrt{1 - x^2}} = \arcsin x + C \), where \( C \) is the constant of integration.
Verify the result by differentiating \( \arcsin x + C \) to confirm it returns the original integrand \( \frac{1}{\sqrt{1 - x^2}} \).

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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 expressions like 1 - x² by substituting x with a trigonometric function such as sin(θ). This transforms the integral into a trigonometric integral that is easier to evaluate.
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Introduction to Trigonometric Functions

Inverse Trigonometric Functions

Integrals involving expressions like 1/√(1 - x²) often result in inverse trigonometric functions, specifically arcsin(x). Recognizing this form helps directly identify the antiderivative without complex manipulation.
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Derivatives of Other Inverse Trigonometric Functions

Basic Integration Techniques

Understanding fundamental integration rules and recognizing standard integral forms allows for efficient evaluation. For example, knowing that ∫ dx/√(1 - x²) = arcsin(x) + C avoids unnecessary steps.
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Basic Rules for Definite Integrals
Related Practice
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 0 to ∞ of (dθ / (θ² - 1))

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Textbook Question

Moment about y-axis:

A thin plate of constant density δ = 1 occupies the region enclosed by the curve

y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.

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Textbook Question

Use reduction formulas to evaluate the integrals in Exercises 41–50.

∫ 16x^3 (ln(x))^2 dx

Textbook Question

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (y + 4) / (y² + y) dy from 1/2 to 1

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.

∫ (x² dx) / (4 + x²)

Textbook Question

87. Find the area of the region that lies between the curves y = sec x and y = tan x from x = 0 to x = π/2.