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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.AAE.5
Chapter 8, Problem 8.AAE.5

Evaluate the integrals in Exercises 1–6.
∫ dt / (t - √(1 - t²))

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1
Start by examining the integral: \(\int \frac{dt}{t - \sqrt{1 - t^{2}}}\). Notice the expression in the denominator involves both \(t\) and \(\sqrt{1 - t^{2}}\), which suggests a trigonometric substitution might simplify the square root term.
Use the substitution \(t = \sin(\theta)\), which implies \(dt = \cos(\theta) d\theta\). This substitution is helpful because \(\sqrt{1 - t^{2}}\) becomes \(\sqrt{1 - \sin^{2}(\theta)} = \cos(\theta)\).
Rewrite the integral in terms of \(\theta\): replace \(t\) with \(\sin(\theta)\) and \(dt\) with \(\cos(\theta) d\theta\). The integral becomes \(\int \frac{\cos(\theta) d\theta}{\sin(\theta) - \cos(\theta)}\).
To simplify the integral, consider dividing numerator and denominator by \(\cos(\theta)\) (assuming \(\cos(\theta) \neq 0\)), which transforms the integral into \(\int \frac{d\theta}{\tan(\theta) - 1}\). This form is easier to handle.
Next, use the substitution \(u = \tan(\theta) - 1\), so that \(du = \sec^{2}(\theta) d\theta\). Express \(d\theta\) in terms of \(du\) and rewrite the integral accordingly. This will allow you to integrate with respect to \(u\) and then back-substitute to \(t\).

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Key Concepts

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

Integration Techniques

Integration techniques involve methods such as substitution, partial fractions, or trigonometric substitution to simplify and evaluate integrals. Recognizing the form of the integrand helps determine the appropriate technique to apply for solving the integral efficiently.
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Trigonometric Substitution

Trigonometric substitution is used to simplify integrals involving expressions like √(1 - t²) by substituting t = sin(θ) or t = cos(θ). This transforms the integral into a trigonometric form that is often easier to integrate.
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Introduction to Trigonometric Functions

Algebraic Manipulation of the Integrand

Algebraic manipulation involves rewriting the integrand to a simpler or more recognizable form, such as rationalizing the denominator or combining terms. This step is crucial to make the integral more approachable for standard integration methods.
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Completing the Square to Rewrite the Integrand
Related Practice
Textbook Question

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Let R be the "triangular" region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.

Find the volume of the solid generated by revolving R about

a. the x-axis.

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Textbook Question

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Textbook Question

Finding volume

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