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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1:10 minutes

Problem 4.9.43

Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.

∫ (sec² x - 1) dx

Verified step by step guidance

Step 1: Recognize the integral ∫(sec² x - 1) dx as a sum of two separate terms. You can split the integral into ∫sec² x dx - ∫1 dx.

Step 2: Recall the standard integral formula for sec² x. The integral of sec² x is tan(x) + C, where C is the constant of integration.

Step 3: Recall the standard integral formula for 1. The integral of 1 with respect to x is x + C.

Step 4: Combine the results from Step 2 and Step 3. The indefinite integral becomes tan(x) - x + C, where C is the constant of integration.

Step 5: To check your work, differentiate the result tan(x) - x + C. The derivative of tan(x) is sec² x, and the derivative of -x is -1. Adding these together gives sec² x - 1, which matches the original integrand.

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Key Concepts

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

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antiderivation, where we seek a function F(x) such that F'(x) equals the integrand.

Basic Integration Rules

Basic integration rules are fundamental techniques used to compute integrals. For example, the integral of sec²(x) is a standard result, yielding tan(x) as its antiderivative. Understanding these rules is essential for solving integrals efficiently and accurately, as they provide a foundation for more complex integration techniques.

Verification by Differentiation

Verification by differentiation involves checking the correctness of an indefinite integral by differentiating the result. If the derivative of the antiderivative matches the original integrand, the integration is confirmed to be correct. This step is crucial in calculus to ensure that the integration process has been performed accurately.