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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 8.1.16
Chapter 8, Problem 8.1.16

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 / √(2θ - θ²))

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1
Start by examining the integral: \(\int \frac{d\theta}{\sqrt{2\theta - \theta^{2}}}\). Notice that the expression under the square root is a quadratic in \(\theta\).
Rewrite the quadratic expression inside the square root in a more recognizable form by completing the square: \(2\theta - \theta^{2} = -\left(\theta^{2} - 2\theta\right) = -\left(\theta^{2} - 2\theta + 1 - 1\right) = -\left( (\theta - 1)^{2} - 1 \right) = 1 - (\theta - 1)^{2}\).
Substitute \(x = \theta - 1\) to simplify the integral. Then, \(d\theta = dx\), and the integral becomes \(\int \frac{dx}{\sqrt{1 - x^{2}}}\).
Recognize that the integral \(\int \frac{dx}{\sqrt{1 - x^{2}}}\) is a standard form whose antiderivative is \(\arcsin x + C\).
Finally, substitute back \(x = \theta - 1\) to express the answer in terms of the original variable \(\theta\): the integral equals \(\arcsin(\theta - 1) + C\).

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Key Concepts

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

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a part of the integrand, the integral can often be transformed into a more manageable form. This technique is especially useful when the integrand contains composite functions.
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Substitution With an Extra Variable

Completing the Square

Completing the square rewrites a quadratic expression in the form ax² + bx + c as a perfect square plus or minus a constant. This method helps simplify expressions under square roots, making them easier to integrate, often leading to trigonometric substitutions.
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Completing the Square to Rewrite the Integrand

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots of quadratic polynomials with trigonometric functions. This leverages identities like sin²x + cos²x = 1 to simplify integrals, especially those involving √(a² - x²), √(x² - a²), or √(x² + a²).
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.

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Textbook Question

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

∫₁^∞ (1 / x^(1/5)) dx

Textbook Question

Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.

Textbook Question

What is the largest value that

∫ from a to b x√(2x - x²) dx

can have for any a and b? Give reasons for your answer.

1
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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.

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Textbook Question

Evaluate the integrals in Exercises 1–24 using integration by parts.

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