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

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0:56 minutes

Problem 4.R.99

Textbook Question

90–103. Indefinite integrals Determine the following indefinite integrals.

∫ (12/x)dx

Verified step by step guidance

Step 1: Recognize that the integral ∫ (12/x) dx involves a basic logarithmic rule. Recall that the integral of 1/x with respect to x is ln|x|.

Step 2: Factor out the constant 12 from the integral. This simplifies the expression to 12 ∫ (1/x) dx.

Step 3: Apply the logarithmic integration rule. The integral of 1/x dx is ln|x|, so the expression becomes 12 ln|x|.

Step 4: Add the constant of integration, C, to account for the indefinite nature of the integral. The result is 12 ln|x| + C.

Step 5: Verify the result by differentiating 12 ln|x| + C. The derivative of ln|x| is 1/x, and multiplying by 12 gives back the original integrand, confirming the solution.

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Key Concepts

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

Indefinite Integral

An indefinite integral represents a family of functions whose derivative is the integrand. It is expressed without specific limits and includes a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation.

Integration of Rational Functions

Rational functions are ratios of polynomials. To integrate a rational function, one often uses techniques such as substitution or partial fraction decomposition. In the case of the integral ∫ (12/x)dx, recognizing that this is a simple rational function allows for straightforward integration using the natural logarithm.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base 'e', where 'e' is approximately 2.71828. It is particularly important in calculus because the derivative of ln(x) is 1/x, making it a key function when integrating expressions involving 1/x, such as in the integral ∫ (12/x)dx.