Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.5.12

Chapter 5, Problem 5.5.12

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals
∫(e³ˣ ⁺¹ d𝓍

Verified step by step guidance

Recognize that the integral involves an exponential function with a linear term in the exponent. To simplify, use the substitution

u = 3𝓍 + 1

, where

a = 3

and

b = 1

Differentiate the substitution

u = 3𝓍 + 1

with respect to

𝓍

to find

du/d𝓍 = 3

, or equivalently,

du = 3 d𝓍

Solve for

d𝓍

in terms of

du

d𝓍 = du/3

. Substitute this and

u

into the integral, replacing

3𝓍 + 1

with

u

and

d𝓍

with

du/3

The integral now becomes

∫e^u (du/3)

. Factor out the constant

1/3

to simplify the expression to

(1/3) ∫e^u du

Evaluate the integral of

e^u

, which is simply

e^u

. Replace

u

with the original substitution

3𝓍 + 1

to return to the variable

𝓍

. The final result will include a constant of integration,

C

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Key Concepts

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

Substitution Method in Integration

The substitution method is a technique used in calculus to simplify the process of integration. By substituting a part of the integrand with a new variable, it transforms the integral into a more manageable form. This method is particularly useful when dealing with composite functions, allowing for easier integration by changing the variable of integration.

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits of integration and include a constant of integration, typically denoted as 'C'. Understanding indefinite integrals is crucial for finding antiderivatives and solving differential equations, as they provide the general solution to integration problems.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = e^(kx), where 'e' is the base of natural logarithms and 'k' is a constant. These functions are characterized by their rapid growth and unique properties, such as the fact that their derivative is proportional to the function itself. Recognizing the behavior of exponential functions is essential for evaluating integrals involving them, particularly in the context of substitution.

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Exponential Functions

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∫¹₁/₂ (t⁻³ ― 8) dt

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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) = 1/(𝓍² + 1) on [―1, 1]

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals∫(e³ˣ ⁺¹ d𝓍

Verified video answer for a similar problem:

Key Concepts

Substitution Method in Integration

Indefinite Integrals

Exponential Functions

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals
∫(e³ˣ ⁺¹ d𝓍