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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 8.1.38
Chapter 8, Problem 8.1.38

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.
∫ (dθ / cos θ - 1)

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Rewrite the integral to clarify the expression: consider the integral \( \int \frac{d\theta}{\cos \theta - 1} \).
Recognize that the denominator \( \cos \theta - 1 \) can be tricky to integrate directly, so use a trigonometric identity to simplify it. Recall that \( \cos \theta - 1 = -2 \sin^2 \left( \frac{\theta}{2} \right) \).
Substitute this identity into the integral to get \( \int \frac{d\theta}{-2 \sin^2 \left( \frac{\theta}{2} \right)} = -\frac{1}{2} \int \csc^2 \left( \frac{\theta}{2} \right) d\theta \).
Make a substitution to simplify the integral: let \( u = \frac{\theta}{2} \), so \( d\theta = 2 du \). Substitute into the integral to get \( -\frac{1}{2} \int \csc^2(u) \cdot 2 du = - \int \csc^2(u) du \).
Recall the integral formula \( \int \csc^2 x \, dx = -\cot x + C \). Use this to write the integral in terms of \( u \), then substitute back \( u = \frac{\theta}{2} \) 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.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They simplify integrals by transforming complex expressions into more manageable forms, such as rewriting 1/(cos θ - 1) using identities like the Pythagorean or half-angle formulas.
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Verifying Trig Equations as Identities

Substitution Method

The substitution method involves changing variables to simplify an integral. By letting a new variable represent a function inside the integral, the integral can become easier to evaluate. For example, substituting u = cos θ - 1 can transform the integral into a rational function in terms of u.
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Integration of Rational Functions

Integrating rational functions involves integrating ratios of polynomials or expressions that can be manipulated into polynomial ratios. Techniques include partial fraction decomposition or algebraic manipulation, which help break down complex fractions into simpler terms that are easier to integrate.
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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 4 to ∞ of (dx / (√x - 1))

Textbook Question

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.

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Textbook Question

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (8x² + 8x + 2) / (4x² + 1)² dx

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Textbook Question

Evaluate the integrals in Exercises 1–14.

∫ (2 dx) / √(1 - 4x²) from 0 to 1/(2√2)

Textbook Question

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ cos(θ / 2) cos(7θ) dθ

Textbook Question

Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)

To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.

∫ arctan x dx