Applying basic integration rules:

• The most general form of an antiderivative includes a constant of integration
• The antiderivative of $x^{n}$ includes the term $\frac{x^{n + 1}}{n + 1}$, where $n$ cannot equal -1
• The antiderivative of $\cos(x)$ is $\sin(x) +$$C$.


  True
• The power rule of integration applies only to integer values of $n$.


  False
• The antiderivative of $5x^{2}$ is \frac{5x^{3}}{3} + C
• Steps to find the antiderivative of $x^{3}$


  1️⃣ Increase exponent by 1: $3 +$$1 =$$4$

  2️⃣ Divide by new exponent: $\frac{x^{4}}{4}$

  3️⃣ Add the constant of integration: $\frac{x^{4}}{4} +$$C$
• Steps to find the antiderivative of $5x^{2}$


  1️⃣ Increase the exponent by 1: $2 +$$1 =$$3$

  2️⃣ Divide by the new exponent: $\frac{x^{3}}{3}$

  3️⃣ Multiply by the constant 5: $\frac{5x^{3}}{3}$

  4️⃣ Add the constant of integration: $\frac{5x^{3}}{3} +$$C$
• The Constant Multiple Rule includes a constant of integration because the derivative of any constant is zero.
• The Difference Rule states that $\int [f(x) - g(x)] \, dx =$$\int f(x) \, dx - \int g(x) \, dx +$$C$, where $C$ is the constant of integration.
• The most general form of an antiderivative includes a constant of integration because the derivative of any constant is zero.

  True
• An antiderivative of a function $f(x)$ is a function $F(x)$ such that $F'(x) =$$f(x)$. The most general form of an antiderivative includes a constant of integration
• In the power rule of integration, $n$ can be any real number except -1
• The antiderivative of $\sqrt{x}$ is $\frac{2x^{3 / 2}}{3} +$$C$ because $\sqrt{x}$ can be written as $x^{1 / 2}$ and $\frac{1}{2} +$$1 =$$\frac{3}{2}$.exponent
• The antiderivative of $5x^{2}$ is $\frac{5x^{3}}{3} +$$C$.


  True
• The constant multiple rule of integration states that $\int k \cdot f(x) \, dx =$$k \cdot \int f(x) \, dx +$$< blank_{s}tart > C < blank_{e}nd > < distractors > \frac{1}{k} ||| k ||| 0 < / distractors > < cloze_{e}nd > < truefalse_{s}tart > < line > 65 < / line > < statement_{s}tart > The constant multiple rule requires adding the constant of integration after multiplying by < latex > k$.<statement_end><answer_start>True<answer_end><truefalse_end>

  
  

  <cloze_start>The sum rule of integration states that \int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx + C
• The Sum Rule of integration states that the antiderivative of the sum of two functions is the sum of their antiderivatives.


  True
• What is the antiderivative of 3x^2 - sin(x)?

  x^3 + cos(x) + C
• The constant of integration is added because the derivative of any constant is zero.


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


  $- \cos(x) +$$C$
• If $f(x) =$$2x$, what is its antiderivative $F(x)$?


  $x^{2} +$$C$
• What is the antiderivative of $x^{3}$?


  $\frac{x^{4}}{4} +$$C$
• To find the antiderivative of $5x^{2}$, we first apply the power rule to $x^{2}$.


  True
• The power rule of integration applies to $x^{n}$ where $n$ can be any real number.


  False
• Match the function with its antiderivative:

  $x^{3}$ ↔️ $\frac{x^{4}}{4} +$$C$

  $\sqrt{x}$ ↔️ $\frac{2x^{3 / 2}}{3} +$$C$

  $\frac{1}{x^{2}}$ ↔️ $- \frac{1}{x} +$$C$
• Match the function with its antiderivative using the Constant Multiple Rule:

  $3x^{2}$ ↔️ $x^{3} +$$C$

  $- 2\sin(x)$ ↔️ $- 2\cos(x) +$$C$
• Steps to find the antiderivative of 3x^{2} - \sin(x)</latex>

  1️⃣ Apply the Difference Rule: $\int [3x^{2} - \sin(x)] \, dx =$$\int 3x^{2} \, dx - \int \sin(x) \, dx +$$C$

  2️⃣ Find the antiderivative of $3x^{2}$: $x^{3} +$$C_{1}$

  3️⃣ Find the antiderivative of $\sin(x)$: $- \cos(x) +$$C_{2}$

  4️⃣ Combine the results: $x^{3} +$$\cos(x) +$$C$
• What is the antiderivative of $3x^{2} - \sin(x)$?


  $x^{3} +$$\cos(x) +$$C$
• What is the purpose of the constant of integration $C$ in antiderivatives?


  Derivative of any constant is zero
• What is the antiderivative of $x^{3}$ using the power rule?


  $\frac{x^{4}}{4} +$$C$
• Steps to apply the power rule of integration to x^{n}</latex>:

  1️⃣ Increase the exponent by 1

  2️⃣ Divide by the new exponent

  3️⃣ Add the constant of integration $C$
• The power rule of integration states that the antiderivative of $x^{n}$ is \frac{x^{n + 1}}{n + 1}
• The difference rule of integration involves subtracting the antiderivatives of two functions.

  True
• The antiderivative of 5 is 5x + C
• What is the formula for the Sum Rule of integration?

  $\int [f(x) +$$g(x)] \, dx =$$\int f(x) \, dx +$$\int g(x) \, dx +$$C$
• Steps to find the antiderivative of 3x^2 - sin(x) using the Difference Rule

  1️⃣ Apply the Difference Rule to split the integral

  2️⃣ Apply the Constant Multiple Rule and Power Rule to 3x^2

  3️⃣ Find the antiderivative of sin(x)

  4️⃣ Combine the antiderivatives and add C
• The simplified antiderivative of x^2 - 4x + 5 is \frac{x^{3}}{3} - 2x^{2} + 5x + C
• When rewritten in power form, $\sqrt{x} +$$\frac{1}{x^{3}}$ becomes x^{1 / 2} + x^{ - 3}
• What is the antiderivative of $f(x) =$$5$?


  5x + C
• The antiderivative of a constant function $f(x) =$$k$ is a linear function of the form $F(x) =$$kx +$$C$.
• Steps to find the antiderivative of $f(x) =$$5$


  1️⃣ Apply the power rule: $\int 5 \, dx =$$5 \int x^{0} \, dx$

  2️⃣ Increase the exponent by 1: $0 +$$1 =$$1$

  3️⃣ Divide by the new exponent: $\frac{5x}{1} =$$5x$

  4️⃣ Add the constant of integration: $5x +$$C$