6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

• An antiderivative of a function $f(x)$ is a function $F(x)$ such that when you differentiate $F(x)$, you get back $f(x)$. Mathematically, $F'(x) =$$f(x)$.derivative
• The process of finding an antiderivative is called integration.
• Match the components of the indefinite integral notation with their meanings:

  $\int$ ↔️ Integral symbol

  $f(x)$ ↔️ Integrand

  $dx$ ↔️ Variable of integration

  $F(x)$ ↔️ Antiderivative
• Why is the constant of integration $C$ added to the antiderivative?


  Derivative of a constant is zero
• The antiderivative of $3x^{2}$ is $x^{3} +$$C$ because $\frac{d}{dx}(x^{3} +$$C) =$$3x^{2}$.
• What is the power rule for integration?

  \int x^{n} \, dx = \frac{x^{n + 1}}{n + 1} + C \quad (n \neq - 1)</latex>
• The constant multiple rule states that $\int kf(x) \, dx =$$k \int f(x) \, dx$, where k is a constant
• The antiderivative of a constant $k$ is $kx +$$C$.
• What is the antiderivative of $\sin(x)$?


  $- \cos(x) +$$C$
• Match the integration components with their symbols:

  Integral symbol ↔️ $\int$

  Integrand ↔️ $f(x)$

  Variable of integration ↔️ $dx$

  Constant of integration ↔️ $C$
• The antiderivative of $2x$ is x^2 + C</latex>.
• What is the power rule for integration when $n =$$0$?


  $\int x^{0} \, dx =$$x +$$C$
• The antiderivative of $\frac{1}{x}$ is \ln |x| + C</latex>
• Match the components of the indefinite integral with their meanings:

  Integral symbol ↔️ Indicates integration process

  Integrand ↔️ Function being integrated

  Variable of integration ↔️ Indicates the variable
• What is the antiderivative of $\cos(x)$?


  $\sin(x) +$$C$
• \int e^{x} \, dx = e^{x}
• What is the antiderivative of $\frac{1}{x}$?


  $\ln |x| +$$C$
• The indefinite integral represents all possible antiderivatives of a function.
• \int f(x) \, dx = F(x) + C
• What does the constant of integration $C$ represent in an indefinite integral?


  Possible constant terms
• Steps to find the indefinite integral of $3x^{2}$


  1️⃣ $\int 3x^{2} \, dx$

  2️⃣ $3 \int x^{2} \, dx$

  3️⃣ $3 \cdot \frac{x^{3}}{3} +$$C$

  4️⃣ $x^{3} +$$C$
• The power rule states that $\int x^{n} \, dx =$$\frac{x^{n + 1}}{n + 1} +$$C$ for n \neq - 1</latex>.
• Match the trigonometric function with its antiderivative:

  $\sin(x)$ ↔️ $- \cos(x) +$$C$

  $\cos(x)$ ↔️ $\sin(x) +$$C$
• What is the antiderivative of $2\sin(x) +$$3\cos(x)$?


  $- 2\cos(x) +$$3\sin(x) +$$C$
• An antiderivative F(x)</latex> satisfies the condition $F'(x) =$$f(x)$.
• What is the antiderivative of $3x^{2}$?


  $x^{3} +$$C$
• \int f(x) \, dx = F(x) + C
• What is integration the process of finding?

  An antiderivative
• The symbol used to indicate integration is \int
• The integrand is the function being integrated.
• What is the variable with respect to which integration is performed called?

  Variable of integration
• The constant added to account for potential constant terms in integration is called the constant of integration
• The Power Rule states that $\int x^{n} \, dx =$$\frac{x^{n + 1}}{n + 1} +$$C$ for $n \neq - 1$.
• For what condition on $n$ is the Power Rule valid?


  $n \neq - 1$
• The Constant Multiple Rule states that \int kf(x) \, dx = k \int f(x) \, dx</latex>, where $k$ is a constant
• What does the Sum/Difference Rule allow you to do with the integral of $f(x) \pm g(x)$?


  Split into two integrals
• The antiderivative of a constant $k$ is $kx +$$C$.
• What is the antiderivative of $\int 3 \, dx$?


  3x + C</latex>
• What does the Constant Multiple Rule state for finding antiderivatives?

  $\int kf(x) \, dx =$$k \int f(x) \, dx$
• The Sum/Difference Rule allows you to split the integral of a sum or difference into separate integrals.