Applying basic integration rules:

• What is the antiderivative of x^{3}</latex>?

  $\frac{x^{4}}{4} +$$C$
• To integrate $5x^{2}$, we can factor out the constant 5
• The power rule of integration includes a constant of integration
• What is the antiderivative of $3x^{2}$?


  $x^{3} +$$C$
• Steps to integrate $\int \sin(2x) \, dx$


  1️⃣ Let $u =$$2x$

  2️⃣ Find $du =$$2 \, dx$

  3️⃣ Solve for $dx =$$\frac{du}{2}$

  4️⃣ Substitute into the integral

  5️⃣ Integrate to get $- \frac{1}{2} \cos(u) +$$C$

  6️⃣ Replace $u$ with $2x$

  7️⃣ Simplify to $- \frac{1}{2} \cos(2x) +$$C$
• What is the integral of $2e^{x}$?


  $2e^{x} +$$C$
• The constant multiple rule states $\int k \cdot f(x) \, dx =$$k \int f(x) \, dx$.


  True
• Match the trigonometric function with its integral:

  $\sin(x)$ ↔️ $- \cos(x) +$$C$

  $\cos(x)$ ↔️ $\sin(x) +$$C$

  $\tan(x)$ ↔️ $\ln|\sec(x)| +$$C$

  $\cot(x)$ ↔️ $\ln|\sin(x)| +$$C$
• Steps to integrate \int \sin(3x) \, dx</latex>

  1️⃣ Let $u =$$3x$, then $du =$$3 \, dx$ and $dx =$$\frac{du}{3}$

  2️⃣ Rewrite the integral: $\frac{1}{3} \int \sin(u) \, du$

  3️⃣ Integrate: $- \frac{1}{3} \cos(u) +$$C$

  4️⃣ Substitute back: $- \frac{1}{3} \cos(3x) +$$C$
• What is the integral of $\tan(x)$?


  $\ln|\sec(x)| +$$C$
• Steps to integrate $\tan(x)$


  1️⃣ Rewrite $\tan(x) =$$\frac{\sin(x)}{\cos(x)}$

  2️⃣ Let $u =$$\cos(x)$, so $du =$$- \sin(x) \, dx$

  3️⃣ Rewrite the integral as $- \int \frac{1}{u} \, du$

  4️⃣ Integrate to get $- \ln|u| +$$C$

  5️⃣ Substitute back to $\ln|\sec(x)| +$$C$
• The power rule of integration states that $\int x^{n} \, dx =$$\frac{x^{n + 1}}{n + 1} +$$C$, where $n \neq - 1$ and $C$ is the constant of integration
• What is the constant multiple rule of integration?

  \int k \cdot f(x) \, dx = k \int f(x) \, dx</latex>
• The integral of (x^{2} + 3x)</latex> is $\frac{x^{3}}{3} +$$\frac{3x^{2}}{2} +$$C$.


  True
• The constant multiple rule states that \int k \cdot f(x) \, dx = k \int f(x) \, dx</latex>.


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


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


  $\frac{1}{3} \sin(3x) +$$C$
• What is the integral of $x^{4}$?


  $\frac{x^{5}}{5} +$$C$
• What is the integral of $(x^{2} +$$3x)$?


  $\frac{x^{3}}{3} +$$\frac{3x^{2}}{2} +$$C$
• What is the integral of $\tan(x)$?


  $\ln|\sec(x)| +$$C$
• The integral of $\cos(x)$ is \sin(x)
• The integral of $\sin(u)$ is - \cos(u)
• What is the constant of integration represented by in the integral of $e^{x}$?


  C
• Match the term with its original function:

  $\int e^{x} \, dx$ ↔️ $e^{x} +$$C$

  $\int 3e^{x} \, dx$ ↔️ $3e^{x} +$$C$
• When integrating $\ln(x)$, the expression $x \ln(x) - x$ is part of the antiderivative.


  True
• What is the Constant Multiple Rule in integration?

  $\int k \cdot f(x) \, dx =$$k \int f(x) \, dx$
• What does the Constant Multiple Rule state in integration?

  Constant times integral of function
• Which rule is used to integrate $x^{2}$ in the example?


  Power Rule
• Write the general formula for the Constant Multiple Rule.

  \int k \cdot f(x) \, dx = k \int f(x) \, dx</latex>
• What is the integral of $e^{x}$ without the Constant Multiple Rule?


  $e^{x} +$$C$
• The integral of $3e^{x}$ is $3e^{x} +$$C$


  True
• Steps to integrate $3e^{x}$ using the Constant Multiple Rule and Exponential Rule.


  1️⃣ Apply the Constant Multiple Rule: $\int 3e^{x} \, dx =$$3 \int e^{x} \, dx$

  2️⃣ Integrate $e^{x}$ using the Exponential Rule: $3e^{x} +$$C$
• The Sum and Difference Rule allows you to integrate terms separately.

  True
• Match the function type with the appropriate integration rule:

  Power Function ↔️ Power Rule

  Constant Multiple Function ↔️ Constant Multiple Rule

  Sum/Difference of Functions ↔️ Sum/Difference Rule

  Trigonometric Function ↔️ Trigonometric Integral Rules

  Exponential Function ↔️ Exponential Integration Rule
• $\int \cos(x) \, dx =$$\sin(x) +$$C$ is the integral of a trigonometric function
• The power rule of integration is valid when $n =$$- 1$.


  False
• What does the sum and difference rule of integration state?

  $\int (f(x) \pm g(x)) \, dx =$$\int f(x) \, dx \pm \int g(x) \, dx$
• What is the antiderivative of $x^{4}$?


  $\frac{x^{5}}{5} +$$C$
• The integral of $\sin(x)$ is -\cos(x) + C</latex>, where $C$ is the constant of integration
• When using u-substitution for $\int \sin(2x) \, dx$, $du =$$2 \, dx$.


  True