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.R.41

Chapter 5, Problem 5.R.41

Evaluating integrals Evaluate the following integrals.

∫ (9𝓍⁸―7𝓍⁶) d𝓍

Verified step by step guidance

Step 1: Recognize that the integral ∫ (9𝓍⁸ ― 7𝓍⁶) d𝓍 is a polynomial integral, which can be solved term by term using the power rule for integration.

Step 2: Apply the power rule for integration to the first term, 9𝓍⁸. The power rule states that ∫ 𝓍ⁿ d𝓍 = (𝓍ⁿ⁺¹)/(n+1) + C, where n is the exponent. For 9𝓍⁸, the integral becomes (9𝓍⁹)/9.

Step 3: Apply the power rule for integration to the second term, -7𝓍⁶. Using the same rule, the integral becomes (-7𝓍⁷)/7.

Step 4: Combine the results from Step 2 and Step 3 into a single expression. Remember to include the constant of integration, C, at the end.

Step 5: Simplify the coefficients in the combined expression to finalize the integral in its simplified form.

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Key Concepts

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

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function on a given interval. It is the reverse process of differentiation and can be used to calculate quantities such as total distance, area, and volume. The integral can be definite, with specific limits, or indefinite, representing a family of functions.

Power Rule for Integration

The Power Rule for Integration is a specific technique used to integrate polynomial functions. It states that the integral of x raised to the power n (where n is not equal to -1) is given by (x^(n+1))/(n+1) + C, where C is the constant of integration. This rule simplifies the process of integrating polynomials by allowing for straightforward application to each term.

Constant Multiplication in Integration

When integrating a function that includes a constant multiplied by a variable term, the constant can be factored out of the integral. This means that if you have a function of the form k*f(x), where k is a constant, the integral can be expressed as k*∫f(x)dx. This property simplifies the integration process and allows for easier calculations.

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Additional Rules for Indefinite Integrals

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Textbook Question

Evaluating integrals Evaluate the following integrals.

∫₀^²π cos² 𝓍/6 d𝓍

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Textbook Question

Area of regions Compute the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.

ƒ(𝓍) = 16―𝓍² on [―4, 4]

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Textbook Question

(b) Find the average value of ƒ shown in the figure on the interval [2,6] and then find the point(s) c in (2, 6) guaranteed to exist by the Mean Value Theorem for Integrals.

Textbook Question

Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer.

∫₀² (𝓍²―4) d𝓍

Textbook Question

Function defined by an integral Let ƒ(𝓍) = ∫₀ˣ (t ― 1)¹⁵ (t―2)⁹ dt .

Textbook Question

Geometry of integrals Without evaluating the integrals, explain why the following statement is true for positive integers n:

∫₀¹ 𝓍ⁿd𝓍 + ∫₀¹ ⁿ√(𝓍d𝓍) = 1

Evaluating integrals Evaluate the following integrals. ∫ (9𝓍⁸―7𝓍⁶) d𝓍

Verified video answer for a similar problem:

Key Concepts

Integration

Power Rule for Integration

Constant Multiplication in Integration

Evaluating integrals Evaluate the following integrals.

∫ (9𝓍⁸―7𝓍⁶) d𝓍