Skip to main content
Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.5.23
Chapter 5, Problem 5.5.23

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  
                                                                                                                                                                    
 βˆ« 𝓍³ (𝓍⁴ + 16)⁢ d𝓍

Verified step by step guidance
1
Step 1: Identify the substitution. Since the integral involves a composite function (𝓍⁴ + 16) raised to a power, let u = 𝓍⁴ + 16. This substitution simplifies the integral by reducing the complexity of the expression.
Step 2: Compute the derivative of u with respect to 𝓍. Differentiating u = 𝓍⁴ + 16 gives du/d𝓍 = 4𝓍³. Rearrange to express du in terms of d𝓍: du = 4𝓍³ d𝓍.
Step 3: Rewrite the integral in terms of u. Substitute u = 𝓍⁴ + 16 and du = 4𝓍³ d𝓍 into the original integral. The integral becomes (1/4) ∫ u⁢ du, where the factor of 1/4 accounts for the 4𝓍³ in du.
Step 4: Integrate with respect to u. Use the power rule for integration: ∫ u⁢ du = (u⁷)/7. Thus, the integral becomes (1/4) * (u⁷)/7.
Step 5: Substitute back u = 𝓍⁴ + 16 into the result. Replace u in the expression with the original variable to return to the original terms: (1/28) * (𝓍⁴ + 16)⁷ + C, where C is the constant of integration.

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, and it is fundamental in calculus for solving problems related to area under curves and accumulation functions.
Recommended video:
05:04
Introduction to Indefinite Integrals

Change of Variables

Change of variables, or substitution, is a technique used in integration to simplify the integrand. By substituting a new variable for a function of the original variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with complex expressions, allowing for easier integration and ultimately leading to the correct antiderivative.
Recommended video:
06:35
Changing Geometries

Differentiation Check

Checking work by differentiation involves taking the derivative of the result obtained from an indefinite integral to verify its correctness. If the derivative of the antiderivative matches the original integrand, the solution is confirmed to be accurate. This step is crucial in calculus as it ensures that the integration process was performed correctly and helps identify any potential errors in the calculations.
Recommended video:
05:02
Determining Differentiability Graphically
Related Practice
Textbook Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

                                                                                                                                                                    

 βˆ« 𝓍eΛ£Β² d𝓍

71
views
Textbook Question

Average velocity The velocity in m/s of an object moving along a line over the time interval [0,6] is v (t) = tΒ² + 3t. Find the average velocity of the object over this time interval.

160
views
Textbook Question

Definite integrals from graphs The figure shows the areas of regions bounded by the graph of Ζ’ and the 𝓍-axis. Evaluate the following integrals.



βˆ«β‚αΆœ Ζ’(𝓍) d𝓍

102
views
Textbook Question

A midpoint Riemann sum Approximate the area of the region bounded by the graph of Ζ’(𝓍) = 100 ― xΒ² and the x-axis on [0, 10] with n = 5 subintervals. Use the midpoint of each subinterval to determine the height of each rectangle (see figure).

111
views
Textbook Question

Average height of a wave The surface of a water wave is described by y = 5 (1 + cos 𝓍) , for ― Ο€ ≀ 𝓍 ≀ Ο€, where y = 0 corresponds to a trough of the wave (see figure). Find the average height of the wave above the trough on [ ―π , Ο€] .

65
views
Textbook Question

Areas of regions Find the area of the region bounded by the graph of Ζ’ and the 𝓍-axis on the given interval.


Ζ’(𝓍) = 𝓍³ ― 1 on [―1, 2]

102
views