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AP Calculus AB: 9.1.3 Antiderivatives of Trigonometric and Exponential Functions

Mathematics11 CardsCreated 3 months ago

This content focuses on finding antiderivatives of common trigonometric and exponential functions by reversing differentiation rules. It emphasizes the use of known integration formulas, highlights the relative difficulty of integration compared to differentiation, and introduces key techniques such as the sum rule and power rule for integration.

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Antiderivatives of Trigonometric and Exponential Functions

  • 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.

  • Discover integration formulas by looking at differentiation formulas backwards.

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Key Terms

Term
Definition

Antiderivatives of Trigonometric and Exponential Functions

  • 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 ...

note

  • Here are some antiderivative formulas.

  • Notice that some functions that are easy to
    differentiate are not as easy to integ...

Find f(x) so that f′(x)=−4e^x−6sinx.

−4e^ x + 6 cos x + C

Evaluate the integral. ∫sec^2xdx

tanx + C

Evaluate the integral. ∫sinx dx

− cos x + C

Evaluate the integral: ∫(2sinx+3cosx) dx.

−2 cos x + 3 sin x + C

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AP Calculus AB: 9.1.3 Antiderivatives of Trigonometric and Exponential Functions
TermDefinition

Antiderivatives of Trigonometric and Exponential Functions

  • 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.

  • Discover integration formulas by looking at differentiation formulas backwards.

note

  • Here are some antiderivative formulas.

  • Notice that some functions that are easy to
    differentiate are not as easy to integrate. It is
    generally the case that it is easier to differentiate
    than integrate.

  • To evaluate this indefinite integral, start by applying
    the sum rule.

  • Now you can evaluate the integral of each term
    individually.

  • Remember, when using the power rule for
    integration, you must multiply by the reciprocal of
    the new exponent.

  • You can always check that your answer is correct
    by taking the derivative.

Find f(x) so that f′(x)=−4e^x−6sinx.

−4e^ x + 6 cos x + C

Evaluate the integral. ∫sec^2xdx

tanx + C

Evaluate the integral. ∫sinx dx

− cos x + C

Evaluate the integral: ∫(2sinx+3cosx) dx.

−2 cos x + 3 sin x + C

Evaluate the integral ∫sin2x/cosx dx.

−2 cos x + C

Evaluate: ∫secx(tanx+secx) dx.

sec x + tan x + C

Evaluate the integral: ∫(1+sin^2θcscθ) dθ

θ−cosθ+C

Evaluate the integral:∫3exdx.

3e^x+C

Evaluate:

∫tan^2xdx

tan x − x + C