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 5.5.11

Chapter 5, Problem 5.5.11

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals.
∫(𝓍 + 1)¹² d𝓍

Verified step by step guidance

Step 1: Identify the substitution. Let u = 𝓍 + 1. This substitution simplifies the expression inside the integral.

Step 2: Compute the derivative of u with respect to 𝓍. Since u = 𝓍 + 1, we have du/d𝓍 = 1, or equivalently, du = d𝓍.

Step 3: Rewrite the integral in terms of u. Substituting u = 𝓍 + 1 and du = d𝓍, the integral becomes ∫u¹² du.

Step 4: Apply the power rule for integration. The integral of uⁿ with respect to u is (uⁿ⁺¹)/(n+1) + C, where C is the constant of integration.

Step 5: Substitute back u = 𝓍 + 1 into the result to express the solution in terms of 𝓍. This completes the evaluation of the indefinite integral.

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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 changing the variable of integration. By substituting a new variable, often denoted as 'u', for a function of 'x', the integral can be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand can be expressed in terms of a simpler variable.

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative gives the integrand. They are expressed without limits of integration and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral involves determining the antiderivative of the function, which can often be achieved through various techniques, including substitution.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. In the context of integration, recognizing polynomial forms is crucial, as they can be integrated using straightforward rules. For example, the integral of x^n is (x^(n+1))/(n+1) + C, which simplifies the evaluation of integrals involving polynomials.

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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.

∫ [(√𝓍 + 1)⁴ / 2√𝓍 d𝓍

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Textbook Question

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.

∫₀⁴ √(16― 𝓍² ) d𝓍

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Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫π/₄^³π/⁴ (cot² 𝓍 + 1) d𝓍

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Textbook Question

Variations on the substitution method Evaluate the following integrals.

∫ 𝓍/(√𝓍―4) d𝓍

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Textbook Question

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