Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.4.50
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ 1 / √(x² - 2x + 5) dx
Verified step by step guidance1
Start by completing the square for the expression inside the square root: \(x^{2} - 2x + 5\). To do this, rewrite it as \(\left(x^{2} - 2x + 1\right) + 4\) because \(1\) is the square of half the coefficient of \(x\).
Express the completed square form explicitly: \(\left(x - 1\right)^{2} + 4\). So the integral becomes \(\int \frac{1}{\sqrt{\left(x - 1\right)^{2} + 4}} \, dx\).
Recognize that the integral now has the form \(\int \frac{1}{\sqrt{u^{2} + a^{2}}} \, du\) where \(u = x - 1\) and \(a = 2\). This suggests using the trigonometric substitution \(u = a \tan \theta\), or \(x - 1 = 2 \tan \theta\).
Differentiate the substitution to find \(dx\): since \(x - 1 = 2 \tan \theta\), then \(dx = 2 \sec^{2} \theta \, d\theta\).
Rewrite the integral in terms of \(\theta\) using the substitution and simplify the square root using the identity \(1 + \tan^{2} \theta = \sec^{2} \theta\). Then proceed to integrate with respect to \(\theta\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Completing the Square
Completing the square is a method used to rewrite a quadratic expression in the form ax² + bx + c as a perfect square plus or minus a constant. This technique simplifies the integrand, making it easier to identify the appropriate substitution, especially for integrals involving square roots of quadratic expressions.
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Completing the Square to Rewrite the Integrand
Trigonometric Substitution
Trigonometric substitution is a technique used to evaluate integrals involving square roots of quadratic expressions by substituting variables with trigonometric functions. Depending on the form (x² ± a² or a² - x²), substitutions like x = a sec θ or x = a tan θ transform the integral into a trigonometric integral that is easier to solve.
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6:04Introduction to Trigonometric Functions
Integration of Functions Involving Square Roots
Integrals containing square roots of quadratic expressions often require algebraic manipulation and substitution to simplify. Understanding how to handle these integrals involves recognizing patterns, applying completing the square, and using trigonometric identities to rewrite and integrate the function effectively.
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07:01Integrals Involving Natural Logs: Substitution
Related Practice
Textbook Question
Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.
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Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ 5 dx / √(25x² - 9), where x > 3/5
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.
∫ 3 sinh(x/2 + ln 5) dx
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Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (cos(√x))/(√x) dx
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫ sin⁴(2x) cos(2x) dx