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

Antiderivatives

Calculus

7. Antiderivatives & Indefinite Integrals

Antiderivatives

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2:56 minutes

Problem 4.9.15

Textbook Question

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.

p(x) = 3 sec² x

Verified step by step guidance

Step 1: Recall the definition of an antiderivative. The antiderivative of a function is a function whose derivative is the original function. For this problem, we need to find the antiderivative of p(x) = 3 sec²(x).

Step 2: Identify the basic antiderivative rule that applies here. The derivative of tan(x) is sec²(x), so the antiderivative of sec²(x) is tan(x). Use this rule to find the antiderivative of 3 sec²(x).

Step 3: Multiply the constant 3 by the antiderivative of sec²(x). This gives us 3 tan(x) as part of the solution.

Step 4: Add the constant of integration, C, to account for all possible antiderivatives. The general antiderivative of p(x) = 3 sec²(x) is therefore 3 tan(x) + C.

Step 5: Verify your work by taking the derivative of the antiderivative you found. Differentiate 3 tan(x) + C, and confirm that the result is 3 sec²(x), which matches the original function.

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Key Concepts

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

Antiderivatives

An antiderivative of a function is another function whose derivative is the original function. In calculus, finding antiderivatives is essential for solving problems related to integration. The process involves determining a function F(x) such that F'(x) equals the given function p(x). Antiderivatives are not unique; they can differ by a constant, represented as F(x) + C, where C is any constant.

Integration

Integration is the process of finding the integral of a function, which is closely related to finding antiderivatives. It can be thought of as the reverse operation of differentiation. The integral of a function over an interval gives the area under the curve of that function. In this context, the integral of p(x) = 3 sec² x will yield the family of functions that are antiderivatives of p(x).

Derivative Check

To verify that a function is indeed an antiderivative, one can take its derivative and check if it matches the original function. This process is crucial in confirming the correctness of the antiderivative found. For example, if F(x) is an antiderivative of p(x), then differentiating F(x) should yield p(x). This step ensures that the integration process was performed correctly.