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4.10 Antiderivatives

2 min read•june 24, 2024

Antiderivatives and indefinite integrals are key concepts in calculus. They represent the opposite of differentiation, allowing us to find functions whose derivatives are known. This process is crucial for solving various mathematical and real-world problems.

Mastering antiderivatives involves understanding notation, applying integration rules, and solving initial-value problems. These skills form the foundation for more advanced integration techniques and applications in calculus and beyond.

Antiderivatives and Indefinite Integrals

General antiderivatives of functions

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  • Function F(x)F(x) is an of f(x)f(x) if F′(x)=f(x)F'(x) = f(x)
    • Antiderivatives "undo" the process of differentiation (the inverse operation of taking a derivative)
    • Example: If f(x)=3x2f(x) = 3x^2, then one antiderivative is F(x)=x3F(x) = x^3
  • General antiderivative includes an arbitrary constant CC representing a family of functions differing by vertical shifts
    • Example: For f(x)=2xf(x) = 2x, the general antiderivative is F(x)=x2+CF(x) = x^2 + C
  • Constant CC determines the vertical position of the antiderivative graph
    • Value of CC can be found using initial or boundary conditions

Indefinite integrals and notation

  • Indefinite integrals denote the general antiderivative of a function
    • Integral symbol ∫\int represents the operation of finding the antiderivative
    • Integrand f(x)f(x) is the function being integrated
    • dxdx indicates the variable of integration
    • Result is the general antiderivative F(x)+CF(x) + C
  • Notation: ∫f(x) dx=F(x)+C\int f(x) \, dx = F(x) + C
    • Example: ∫3x2 dx=x3+C\int 3x^2 \, dx = x^3 + C
  • represents a family of functions, not a specific value
    • The constant CC is also known as the constant of integration

Power rule for integration

  • Power rule: For f(x)=xnf(x) = x^n where n≠−1n \neq -1, ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
  • Steps:
    1. Add 1 to the power of the variable
    2. Divide by the new power
    3. Include the constant of integration CC
  • Example: ∫x4 dx=x55+C\int x^4 \, dx = \frac{x^5}{5} + C
  • Power rule applies to functions with multiple terms by integrating each term separately
    • Example: ∫(3x2+2x) dx=x3+x2+C\int (3x^2 + 2x) \, dx = x^3 + x^2 + C

Initial-value problems via antidifferentiation

  • Initial-value problems involve finding a specific antiderivative satisfying a given initial condition
  • Steps:
    1. Find the general antiderivative using integration techniques (power rule)
    2. Determine the value of constant CC using the initial condition
    3. Substitute CC into the general antiderivative to obtain the specific solution
  • Example: Given f(x)=4x3f(x) = 4x^3 and f(1)=2f(1) = 2, find the specific antiderivative
    1. ∫4x3 dx=x4+C\int 4x^3 \, dx = x^4 + C
    2. f(1)=2  ⟹  14+C=2  ⟹  C=1f(1) = 2 \implies 1^4 + C = 2 \implies C = 1
    3. Specific antiderivative: f(x)=x4+1f(x) = x^4 + 1

Relationship between derivatives and antiderivatives

  • Integration is the process of finding antiderivatives
  • An antiderivative is also called a primitive function
  • The Fundamental Theorem of Calculus connects differentiation and integration as inverse operations

Key Terms to Review (6)

Antiderivative: An antiderivative of a function $f(x)$ is another function $F(x)$ such that the derivative of $F(x)$ is equal to $f(x)$. It is also known as the indefinite integral of $f(x)$.
Exponential Function: An exponential function is a mathematical function where the variable appears as the exponent. These functions exhibit a characteristic pattern of growth or decay, making them important in various fields of study, including calculus, physics, and finance.
Indefinite integral: An indefinite integral, also known as an antiderivative, is a function whose derivative is the given function. It represents a family of functions that differ by a constant.
Initial-value problem: An initial-value problem is a differential equation accompanied by specified values of the unknown function at a given point, called the initial conditions. Solving it involves finding a function that satisfies both the differential equation and the initial conditions.
Natural exponential function: The natural exponential function is defined as $e^x$, where $e$ is Euler's number, approximately equal to 2.71828. It is a fundamental function in calculus with unique properties related to growth and decay.
Trigonometric Functions: Trigonometric functions are mathematical functions that describe the relationships between the sides and angles of a right triangle. They are widely used in various fields, including calculus, to analyze periodic phenomena and model real-world situations.
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