Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.5.20

Chapter 5, Problem 5.5.20

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

∫ [(√𝓍 + 1)⁴ / 2√𝓍 d𝓍

Verified step by step guidance

Step 1: Identify the substitution that simplifies the integral. Let u = √𝓍 + 1. Then, differentiate u with respect to 𝓍 to find du/d𝓍. Since u = √𝓍 + 1, we have du/d𝓍 = 1/(2√𝓍). Rearrange to express d𝓍 in terms of du: d𝓍 = 2√𝓍 du.

Step 2: Rewrite the integral in terms of u. Substitute √𝓍 + 1 with u and d𝓍 with 2√𝓍 du. The integral becomes ∫ [(u⁴) / (2√𝓍)] * (2√𝓍 du). Notice that the √𝓍 terms cancel out, simplifying the integral to ∫ u⁴ du.

Step 3: Integrate u⁴ with respect to u. Use the power rule for integration: ∫ u⁴ du = (u⁵)/5 + C, where C is the constant of integration.

Step 4: Substitute back the original variable 𝓍 into the solution. Recall that u = √𝓍 + 1, so replace u in the result with √𝓍 + 1. The integral becomes ((√𝓍 + 1)⁵)/5 + C.

Step 5: Verify the solution by differentiating the result. Differentiate ((√𝓍 + 1)⁵)/5 + C with respect to 𝓍 and confirm that it simplifies back to the original integrand [(√𝓍 + 1)⁴ / 2√𝓍]. This ensures the solution is correct.

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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.

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.

Differentiation Check

Checking work by differentiation involves taking the derivative of the result obtained from an indefinite integral to verify its correctness. This process ensures that the antiderivative found corresponds to the original integrand. If the derivative of the antiderivative matches the integrand, it confirms that the integration was performed correctly, reinforcing the relationship between differentiation and integration.

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Textbook Question

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∫₁³ ( 2ˣ / 2ˣ + 4 ) d𝓍

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Textbook Question

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ƒ(𝓍) = 𝓍ⁿ on [0,1] , for any positive integer n

Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 𝓍³ on [―1, 1]

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Textbook Question

Area versus net area Graph the following functions. Then use geometry (not Riemann sums) to find the area and the net area of the region described.

The region between the graph of y = 1 - |x| and the x-axis, for -2 ≤ x ≤ 2

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Textbook Question

A midpoint Riemann sum Approximate the area of the region bounded by the graph of ƒ(𝓍) = 100 ― x² 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).

Textbook Question

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫₋π/₄^π/⁴ sec² x dx

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. ∫ [(√𝓍 + 1)⁴ / 2√𝓍 d𝓍

Verified video answer for a similar problem:

Key Concepts

Indefinite Integrals

Change of Variables

Differentiation Check

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

∫ [(√𝓍 + 1)⁴ / 2√𝓍 d𝓍