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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.42
Chapter 8, Problem 8.AAE.42

Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.
∫ csc θ dθ

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1
Recognize that the integral involves the cosecant function, which can be tricky to integrate directly. The substitution \(z = \tan\left(\frac{\theta}{2}\right)\) is a common technique called the Weierstrass substitution, useful for trigonometric integrals.
Express \(\sin \theta\) and \(d\theta\) in terms of \(z\). Recall the half-angle formulas: \(\sin \theta = \frac{2z}{1 + z^2}\) and \(d\theta = \frac{2}{1 + z^2} dz\).
Rewrite the integral \(\int \csc \theta \, d\theta\) as \(\int \frac{1}{\sin \theta} d\theta\). Substitute the expressions for \(\sin \theta\) and \(d\theta\) in terms of \(z\) to transform the integral into one involving \(z\) only.
Simplify the resulting integral in terms of \(z\). This will typically reduce to a rational function of \(z\), which is easier to integrate using standard calculus techniques.
Integrate with respect to \(z\), then substitute back \(z = \tan\left(\frac{\theta}{2}\right)\) to express the answer in terms of \(\theta\).

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Key Concepts

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

Weierstrass Substitution (t = tan(θ/2))

The substitution t = tan(θ/2) transforms trigonometric integrals into rational functions of t, simplifying integration. It uses half-angle identities to rewrite sine and cosine in terms of t, making complex trigonometric integrals more manageable.
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Indefinite Integrals Example 2

Integration of Cosecant Function

The integral of csc θ involves recognizing its standard antiderivative or transforming it using substitutions. Understanding how to rewrite csc θ in terms of sine and applying substitution techniques is essential for evaluating the integral.
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Example 6: Integral of Secant & Cosecant

Trigonometric Identities and Half-Angle Formulas

Half-angle formulas express sine and cosine in terms of tan(θ/2), enabling substitution methods. Mastery of these identities is crucial to convert the integral into a rational function and to back-substitute after integration.
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Solve Trig Equations Using Identity Substitutions
Related Practice
Textbook Question

Finding volume

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.

Textbook Question

2. When applying the formula for integration by parts, how do you choose the u and dv? How can you apply integration by parts to an integral of the form ∫ f(x) dx?

Textbook Question

Evaluate the limits in Exercise 9 and 10 by identifying them with definite integrals and evaluating the integrals.

lim (n → ∞) Σ (from k=1 to n) ln √(1 + k/n)

Textbook Question

Finding volume

The infinite region bounded by the coordinate axes and the curve y = −ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.

1
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Textbook Question

7. What is the goal of the method of partial fractions?

Textbook Question

Finding volume

The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.