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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 5.5.8
Chapter 5, Problem 5.5.8

Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.                                                                                              
                                                                                                                                                                                       
 βˆ« 8𝓍 cos (4𝓍² + 3) d𝓍, u = 4𝓍² + 3

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Step 1: Identify the substitution variable. The problem suggests using u = 4𝓍² + 3. This substitution simplifies the integral by replacing the complex expression inside the cosine function.
Step 2: Compute the derivative of u with respect to 𝓍 to find du. Differentiating u = 4𝓍² + 3 gives du/d𝓍 = 8𝓍. Therefore, du = 8𝓍 d𝓍.
Step 3: Rewrite the integral in terms of u. Substitute u = 4𝓍² + 3 and du = 8𝓍 d𝓍 into the integral. The integral becomes ∫ cos(u) du.
Step 4: Solve the simplified integral. The integral of cos(u) with respect to u is sin(u) + C, where C is the constant of integration.
Step 5: Substitute back the original variable. Replace u with 4𝓍² + 3 to express the solution in terms of 𝓍. The final result is sin(4𝓍² + 3) + C.

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Key Concepts

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

Substitution Method

The substitution method is a technique used in integration to simplify the process by replacing a complex expression with a single variable. In this case, we let u = 4xΒ² + 3, which transforms the integral into a more manageable form. This method is particularly useful when dealing with integrals involving composite functions, as it allows for easier integration and ultimately leads to the solution.
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Differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function changes as its input changes. In the context of checking the result of an integral, differentiating the antiderivative obtained from the integration process should yield the original integrand. This step is crucial for verifying the correctness of the integration and ensuring that no mistakes were made during the substitution or integration steps.
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Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed with a constant of integration (C) because there are infinitely many antiderivatives differing by a constant. Understanding indefinite integrals is essential for solving problems in calculus, as they provide the foundational concept for finding areas under curves and solving differential equations.
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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« sec 4w tan 4w dw

Textbook Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

βˆ«Β²β‚‹β‚‚ [(xΒ³ ― 4x) / (xΒ² + 1)] dx 

Textbook Question

Derivatives of integrals Simplify the following expressions.


d/dt βˆ«β‚€α΅— d𝓍/(1 + 𝓍²) + βˆ«β‚ΒΉ/α΅— dx/(1 + 𝓍²)

Textbook Question

Left and right Riemann sums Use the figures to calculate the left and right Riemann sums for f on the given interval and for the given value of n.

Ζ’(𝓍) = x + 1 on [1,6] ; n = 5

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Textbook Question

Approximating area from a graph Approximate the area of the region bounded by the graph (see figure) and the 𝓍-axis by dividing the interval [1, 7] into n = 6 subintervals. Use a left and right Riemann sum to obtain two different approximations.                                                                                                                                                                         

Textbook Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

βˆ«Β²β°β°β‚‹β‚‚β‚€β‚€ 2x⁡ dx