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AP Calculus AB: 9.1.2 Antiderivatives of Powers

Mathematics9 CardsCreated 3 months ago

This content explains how to find antiderivatives of functions involving powers of π‘₯, emphasizing the concept of indefinite integrals and the constant of integration 𝐢. It highlights the idea that functions can have infinitely many antiderivatives and includes example problems demonstrating how to evaluate integrals involving powers and rational expressions.

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Antiderivatives of Powers of x

  • Given two functions, f and F, F is an antiderivative of f if F (x) = f(x). Antidifferentiation is a process or operation that reverses differentiation.

  • If a function has an antiderivative, then it has an infinite number of antiderivatives. The most general antiderivative includes an arbitrary constant of integration.

  • The properties of antidifferentiation mirror the properties of differentiation.

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Key Terms

Term
Definition

Antiderivatives of Powers of x

  • Given two functions, f and F, F is an antiderivative of f if F (x) = f(x). Antidifferentiation is a process or operation that reverses diff...

note

  • This expression is called an indefinite integral. The integral symbol instructs you to find the most general antiderivative.

Evaluate:

βˆ«βŽ›βŽ10/x^2βˆ’3/x+2⎞⎠ dx

βˆ’10/xβˆ’3ln|x|+2x+C

Evaluate:

∫2/x^5 dx

βˆ’1/2x^4+C

Evaluate:

∫(4xβˆ’3)(2xβˆ’1)dx.

8/3x^3βˆ’5x^2+3x+C

Find the antiderivative of 10x + 2.

5xβ€Š^2 + 2x + C

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AP Calculus AB: 9.1.2 Antiderivatives of Powers
TermDefinition

Antiderivatives of Powers of x

  • Given two functions, f and F, F is an antiderivative of f if F (x) = f(x). Antidifferentiation is a process or operation that reverses differentiation.

  • If a function has an antiderivative, then it has an infinite number of antiderivatives. The most general antiderivative includes an arbitrary constant of integration.

  • The properties of antidifferentiation mirror the properties of differentiation.

note

  • This expression is called an indefinite integral. The integral symbol instructs you to find the most general antiderivative.

  • Notice that a function can have many different antiderivatives.

  • In fact, you can add any constant to one of the antiderivatives and the result is still an antiderivative of the function.

  • Notice that none of these graphs describes the most general antiderivative. Each one is very specific. To describe the most general antiderivative you must add an arbitrary constant to the antiderivative.

  • C is called the constant of integration.

  • Some of the rules for evaluating integrals are a lot like the rules for finding derivatives.

Evaluate:

βˆ«βŽ›βŽ10/x^2βˆ’3/x+2⎞⎠ dx

βˆ’10/xβˆ’3ln|x|+2x+C

Evaluate:

∫2/x^5 dx

βˆ’1/2x^4+C

Evaluate:

∫(4xβˆ’3)(2xβˆ’1)dx.

8/3x^3βˆ’5x^2+3x+C

Find the antiderivative of 10x + 2.

5xβ€Š^2 + 2x + C

Evaluate. ∫dx

x+C

Evaluate:

∫4x^3+2/x^2 dx

2x^2βˆ’2/x+C

Evaluate:

∫(6x^2βˆ’4x+1) dx.

2xβ€Š^3 βˆ’ 2xβ€Š^2 + x + C