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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.1.76b
Chapter 8, Problem 8.1.76b

76. Different Substitutions
b. Show that ∫(1/√(x - x²)) dx = 2 sin⁻¹√x + C using substitution u = √x

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Step 1: Begin by substituting u = √x. This implies that x = u² and dx = 2u du. Rewrite the integral in terms of u.
Step 2: Substitute x = u² into the expression under the square root: √(x - x²) becomes √(u² - u⁴).
Step 3: Factorize the expression under the square root: u² - u⁴ = u²(1 - u²). Thus, √(u² - u⁴) = u√(1 - u²).
Step 4: Replace dx with 2u du and simplify the integral: ∫(1/√(x - x²)) dx becomes ∫(1/(u√(1 - u²)) * 2u du). Cancel out u in the numerator and denominator.
Step 5: The integral simplifies to ∫(2/√(1 - u²)) du. Recognize this as the standard integral for the arcsine function, ∫(1/√(1 - u²)) du = sin⁻¹(u) + C. Substitute back u = √x to get the final result: 2 sin⁻¹√x + C.

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Key Concepts

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

Substitution in Integration

Substitution is a technique used in integration to simplify the integrand. By replacing a variable with another expression, the integral can often be transformed into a more manageable form. In this case, substituting u = √x allows us to express the integral in terms of u, making it easier to evaluate.
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Substitution With an Extra Variable

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹(x), are used to find angles when given the value of a trigonometric function. In this problem, the result involves sin⁻¹(√x), which indicates that the integral's solution relates to the angle whose sine is √x. Understanding these functions is crucial for interpreting the final result of the integral.
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Derivatives of Other Inverse Trigonometric Functions

Definite vs. Indefinite Integrals

Indefinite integrals represent a family of functions and include a constant of integration (C) in their results. The integral in this question is indefinite, meaning it does not have specified limits of integration. Recognizing the difference between definite and indefinite integrals is essential for correctly interpreting the outcome of integration problems.
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Related Practice
Textbook Question

88. Incorrect Calculation

b. Evaluate ∫(from -1 to 1) dx/x or show that the integral does not exist.

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Textbook Question

Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.

b. Is L concave up or concave down on [0, ∞)?

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Textbook Question

63. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. If m is a positive integer, then ∫[0 to π] sin^m(x) dx = 0.

Textbook Question

Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.

b. Use the substitution x = u² and the fact that ∫ from 0 to ∞ of e^(-u²) du = √(π/2) to show that Γ(1/2) = √π.

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Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

67. Let f(x) = √(x³ + 1).

b. Calculate f''(x).

Textbook Question

68. Different methods

b. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.