Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.9.53
Chapter 4, Problem 4.9.53

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (6/√(4 - 4x²))dx

Verified step by step guidance
1
Recognize that the integral ∫ (6/√(4 - 4x²)) dx resembles the standard form of the arcsine function: ∫ (1/√(a² - u²)) du = arcsin(u/a) + C. Here, we need to rewrite the given integral to match this form.
Factor out the constant 6 from the integral: ∫ (6/√(4 - 4x²)) dx = 6 ∫ (1/√(4 - 4x²)) dx.
Simplify the denominator √(4 - 4x²) by factoring out 4: √(4 - 4x²) = √(4(1 - x²)) = 2√(1 - x²). Substitute this back into the integral: 6 ∫ (1/√(4 - 4x²)) dx = 6 ∫ (1/(2√(1 - x²))) dx.
Simplify further by factoring out the constant 1/2: 6 ∫ (1/(2√(1 - x²))) dx = 3 ∫ (1/√(1 - x²)) dx. Now the integral matches the standard arcsine form, where u = x and a = 1.
Apply the formula for the arcsine integral: ∫ (1/√(1 - x²)) dx = arcsin(x) + C. Therefore, the solution becomes 3 * arcsin(x) + C. Finally, check your work by differentiating the result to ensure it matches the original integrand.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, where we seek a function F(x) such that F'(x) equals the integrand.
Recommended video:
05:04
Introduction to Indefinite Integrals

Substitution Method

The substitution method is a technique used in integration to simplify the integrand by changing variables. This involves selecting a new variable, often denoted as 'u', which is a function of 'x', and rewriting the integral in terms of 'u'. This method is particularly useful when dealing with composite functions or expressions that can be simplified through a change of variables.
Recommended video:
07:33
Euler's Method

Differentiation Check

After finding an indefinite integral, it is essential to verify the result by differentiation. This involves taking the derivative of the antiderivative obtained and checking if it equals the original integrand. This step ensures that the integration process was performed correctly and confirms the validity of the solution.
Recommended video:
05:02
Determining Differentiability Graphically
Related Practice
Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ (1/x√(36x² - 36))dx

Textbook Question

{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.


y = 8/(x² + 4) (Witch of Agnesi)

2
views
Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→0 x csc x

Textbook Question

{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = x/6 - sec x on [0,8]

Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.



∫ (3x⁵ - 5x⁹) dx

Textbook Question

Let ƒ(x) = 2x³ - 6x² + 4x. Use Newton’s method to find x₁ given that x₀ = 1.4. Use the graph of f (see figure) and an appropriate tangent line to illustrate how x₁ is obtained from x₀ . <IMAGE>