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 4.9.25
Chapter 4, Problem 4.9.25

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


∫ (4√x - (4 /√x)) dx

Verified step by step guidance
1
Step 1: Break down the integral into separate terms. The given integral is ∫ (4√x - (4/√x)) dx. Rewrite it as ∫ 4√x dx - ∫ (4/√x) dx to handle each term individually.
Step 2: Express the square root terms in terms of exponents. Recall that √x = x^(1/2) and 1/√x = x^(-1/2). Substitute these into the integral: ∫ 4x^(1/2) dx - ∫ 4x^(-1/2) dx.
Step 3: Apply the power rule for integration to each term. The power rule states that ∫ x^n dx = (x^(n+1))/(n+1) + C, where n ≠ -1. For the first term, integrate 4x^(1/2) using the power rule. For the second term, integrate 4x^(-1/2) using the same rule.
Step 4: Combine the results of the integrations. After applying the power rule, you will have expressions for both terms. Add them together to form the complete indefinite integral.
Step 5: Add the constant of integration, C, to the final result. This constant accounts for the family of antiderivatives. To verify your work, differentiate the result and check if 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:
3m
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

Integration Techniques

To solve integrals, various techniques can be employed, including substitution, integration by parts, and recognizing standard forms. In this case, the integrand consists of terms involving powers of x, which can be integrated using the power rule. The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C, applicable for n ≠ -1.
Recommended video:
06:18
Integration by Parts for Definite Integrals

Verification by Differentiation

After finding an indefinite integral, it is essential to verify the result by differentiating the antiderivative. This process ensures that the derivative of the obtained function returns to the original integrand. This step is crucial for confirming the correctness of the integration process and helps identify any potential errors in the calculation.
Recommended video:
05:53
Finding Differentials
Related Practice
Textbook Question

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


lim_x→π/2⁻ (π/2 - x) sec x

1
views
Textbook Question

Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>

Textbook Question

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


lim_x→ 0 (1 - cos 3x) / 8x²

Textbook Question

{Use of Tech} Write the formula for Newton’s method and use the given initial approximation to compute the approximations x₁ and x₂.


f(x) = e⁻ˣ - x; x₀ = ln 2

Textbook Question

{Use of Tech} Finding intersection points Use Newton’s method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations.


y = 1/x and y = 4 - x²

Textbook Question

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = (x+4)/(4-x)