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

8. Definite Integrals

Substitution

Calculus

8. Definite Integrals

Substitution

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2:19 minutes

Problem 5.R.50

Textbook Question

Evaluating integrals Evaluate the following integrals.

∫₁ᵉ d𝓍 / [𝓍(1 + ln 𝓍)]

Verified step by step guidance

Step 1: Recognize that the integral ∫₁ᵉ d𝓍 / [𝓍(1 + ln 𝓍)] involves a logarithmic function in the denominator, suggesting a substitution method might simplify the problem. Let u = ln(𝓍), which implies that du = d𝓍 / 𝓍.

Step 2: Substitute u = ln(𝓍) into the integral. When 𝓍 = 1, u = ln(1) = 0, and when 𝓍 = e, u = ln(e) = 1. The integral now becomes ∫₀¹ du / (1 + u).

Step 3: Recognize that the integral ∫₀¹ du / (1 + u) is a standard form that can be solved using the natural logarithm function. Specifically, the integral of 1 / (1 + u) is ln|1 + u|.

Step 4: Apply the antiderivative formula to evaluate the integral. The result is ln|1 + u| evaluated from u = 0 to u = 1.

Step 5: Substitute the limits of integration into the antiderivative expression ln|1 + u| to complete the evaluation. Simplify the result to express the final answer.

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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 accumulated area under a curve represented by a function. It is the reverse process of differentiation and can be used to calculate quantities such as areas, volumes, and total accumulated change. The integral symbol (∫) denotes the operation, and definite integrals have specified limits, while indefinite integrals do not.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a crucial function in calculus, particularly in integration and differentiation, as it arises in various contexts, including growth processes and compound interest. Understanding its properties, such as ln(ab) = ln(a) + ln(b), is essential for manipulating expressions involving logarithms.

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing the variable of integration. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative.