6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

• The antiderivative of a function is unique.

  False
• How many solutions exist for the antiderivative of a function?

  Infinitely many
• Match the term with its description:

  Derivative ↔️ Finds the rate of change

  Antiderivative ↔️ Finds the accumulation of change

  Solution to derivative ↔️ Unique

  Solution to antiderivative ↔️ Infinitely many
• The constant multiple rule states that $\int kf(x) dx =$$k \int f(x) dx$


  True
• What is the antiderivative of $\sin(x)$?


  $- \cos(x) +$$C$
• What is the integrand in the indefinite integral \int f(x) dx</latex>?

  $f(x)$
• What is the antiderivative of $\sin(x)$?


  $- \cos(x) +$$C$
• What does `f(x)` represent in the indefinite integral notation $\int f(x) dx =$$F(x) +$$C$?


  The integrand
• The constant of integration reflects that the antiderivative is not unique.

  True
• Match the derivative operation with its inverse operation:

  Derivative ↔️ Antiderivative

  Rate of change ↔️ Accumulation of change
• The constant multiple rule allows you to move a constant outside the integral.


  True
• The constant `C` in the indefinite integral accounts for the non-uniqueness of antiderivatives.


  True
• What is the antiderivative of $x^{n}$ using the power rule?


  $\frac{x^{n + 1}}{n + 1} +$$C$
• The power rule for integration states that \int x^{n} dx = \frac{x^{n + 1}}{n + 1} + C
• The sum/difference rule for integration allows you to integrate each term separately.


  True
• What is the result of $\int x^{2} dx$?


  $\frac{x^{3}}{3} +$$C$
• The exponential function rule for integration states that $\int e^{x} dx =$$e^{x} +$$C$.


  True
• What is the result of \int x^{n} dx</latex>?

  $\frac{x^{n + 1}}{n + 1} +$$C$
• What is the result of \int x^{2} dx</latex>?

  $\frac{x^{3}}{3} +$$C$
• What is the result of $\int \sin(x) dx$?


  $- \cos(x) +$$C$
• What is an antiderivative of a function `f(x)`?

  A function `F(x)` such that `F'(x) = f(x)`
• What does finding the derivative of a function find?

  The rate of change
• The antiderivative `F(x)` is the inverse operation of the derivative
• What is the antiderivative of $x^{n}$ according to the power rule?


  $\frac{x^{n + 1}}{n + 1} +$$C$
• What is the antiderivative of $e^{x}$?


  $e^{x} +$$C$
• The indefinite integral is denoted as \int f(x) dx = F(x) + C
• The term `F(x)` in the indefinite integral represents the antiderivative of `f(x)`.

  True
• The indefinite integral is denoted as \int
• What does `dx` indicate in the indefinite integral notation?

  Integration with respect to x
• An antiderivative of a function `f(x)` is a function `F(x)` whose derivative is f(x)
• The antiderivative of a function has infinitely many solutions differing by a constant
• What is the antiderivative of $x^{n}$ using the power rule?


  \frac{x^{n + 1}}{n + 1} + C</latex>
• Match the trigonometric function with its antiderivative:

  \sin(x) ↔️ -\cos(x) + C

  \cos(x) ↔️ \sin(x) + C
• The indefinite integral represents the set of all antiderivatives
• The constant multiple rule states that \int kf(x) dx = k \int f(x)
• What is the constant multiple rule for integration?

  \int kf(x) dx = k \int f(x) dx</latex>
• The sum/difference rule for integration states that \int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx + C
• What is the result of \int 5x^{2} dx</latex>?

  $\frac{5x^{3}}{3} +$$C$
• The exponential function rule for integration states that \int e^{x} dx = e^{x} + C
• Match the integration rule with its result:

  $\int e^{x} dx$ ↔️ $e^{x} +$$C$

  $\int \sin(x) dx$ ↔️ $- \cos(x) +$$C$

  $\int \cos(x) dx$ ↔️ $\sin(x) +$$C$