Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Previous problemNext problem

2:09 minutes

Problem 5.5.12

Textbook Question

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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Related Practice

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. Using the substitution u = tan(x) in ∫ (tan²x / (tan x - 1)) dx leads to ∫ (u² / (u - 1)) du.

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Textbook Question

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

e. The best approach to evaluating ∫(x³ + 1)/(3x²) dx is to use the change of variables u = x³ + 1.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(g) ∫ ƒ' (g(𝓍))g' (𝓍) d(𝓍) = ƒ(g(𝓍)) + C .

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Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.

(a) ∫ e¹⁰ˣ d𝓍

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Textbook Question

Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals.

∫(𝓍 + 1)¹² d𝓍

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Textbook Question

Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.