Understanding properties of definite integrals:

• In the definite integral formula, $b$ represents the upper
• Steps to apply the linear property of definite integrals

  1️⃣ Split the integral into two separate integrals

  2️⃣ Evaluate each integral separately

  3️⃣ Add the results together
• The constant multiple property of definite integrals is given by the formula $\int_{a}^{b} cf(x) dx =$$c\int_{a}^{b} f(x) dx$, where $c$ is a constant
• The constant multiple property simplifies integration by factoring out the constant.


  True
• Evaluate $\int_{0}^{1} (x + 2) dx$ using the sum property.


  $\frac{5}{2}$
• What is the formula for the linear property of definite integrals?

  \int_{a}^{b} [f(x) + g(x)] dx = \int_{a}^{b} f(x) dx + \int_{a}^{b} g(x) dx</latex>
• The interpretation of definite integrals is that they represent the total area
• Match the property with its description:

  Constant Multiple Property ↔️ Allows factoring out a constant

  Linearity Property ↔️ Splits a sum of integrals
• The integration over adjacent intervals property uses the sum of integrals over subintervals
• Match the type of integral with its property:

  Definite Integral ↔️ $\int_{a}^{b} c dx =$$c(b - a)$

  Indefinite Integral ↔️ $\int c dx =$$cx +$$C$
• What is the sum and difference property of definite integrals?

  $\int_{a}^{b} [f(x) \pm g(x)] dx =$$\int_{a}^{b} f(x) dx \pm \int_{a}^{b} g(x) dx$
• In the definite integral formula, $a$ represents the lower
• The linear property of definite integrals states that \int_{a}^{b} [f(x) + g(x)] dx = \int_{a}^{b} f(x) dx + \int_{a}^{b} g(x) dx</latex>sum
• Match the property with its description:

  Definite Integral ↔️ Calculates area under a curve

  Indefinite Integral ↔️ Represents a family of antiderivatives
• What is the formula for the constant multiple property of definite integrals?

  $\int_{a}^{b} cf(x) dx =$$c\int_{a}^{b} f(x) dx$
• The sum and difference property separates functions within integration bounds.

  True
• The result of a definite integral is a numeric value representing area.

  True
• A definite integral calculates the area under a curve between two specified points
• The result of a definite integral is a numeric value representing area.


  True
• The linear property of definite integrals allows breaking down complex integrals into simpler parts.


  True
• The constant multiple property applies to both definite and indefinite integrals.


  True
• Evaluate $\int_{1}^{3} 2x dx$ using the constant multiple property.


  8
• A definite integral calculates the area under a curve between two specified points
• The linear property of definite integrals states that the definite integral of a sum is equal to the sum of the individual definite integrals
• What is the main advantage of the linear property for evaluating definite integrals?

  Simplifies calculations
• Evaluate \int_{1}^{3} 2x dx</latex> using the constant multiple property.

  8
• Steps to evaluate $\int_{0}^{1} (x + 2) dx$ using the sum property:


  1️⃣ Split the integral into $\int_{0}^{1} x dx +$$\int_{0}^{1} 2 dx$

  2️⃣ Evaluate $\int_{0}^{1} x dx =$$\left[\frac{1}{2}x^{2}\right]_{0}^{1} =$$\frac{1}{2}$

  3️⃣ Evaluate $\int_{0}^{1} 2 dx =$$\left[2x\right]_{0}^{1} =$$2$

  4️⃣ Add the results: $\frac{1}{2} +$$2 =$$\frac{5}{2}$
• Splitting an integral at $c =$$1$ in \int_{0}^{3} x dx</latex> requires $\int_{0}^{1} x dx +$$\int_{1}^{3} x dx$.


  True
• What is the value of $\int_{2}^{5} 3 dx$?


  9
• The property of integrating over adjacent intervals states that \int_{a}^{b} f(x) dx = \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx</latex>.


  True
• The formula for a definite integral is \int_{a}^{b} f(x) dx = F(b) - F(a)</latex>


  True
• Definite integrals do not require a constant of integration
• The formula for a definite integral is $\int_{a}^{b} f(x) dx =$$F(b) - F(a)$, where $F(x)$ is the antiderivative
• The constant multiple property states that the definite integral of a function multiplied by a constant is equal to the constant multiplied by the definite integral
• The sum and difference property allows you to split a single definite integral into multiple integrals
• Match the term with its definition:

  $a$ ↔️ Lower limit of integration

  $b$ ↔️ Upper limit of integration

  $f(x)$ ↔️ Integrand

  $F(x)$ ↔️ Antiderivative
• What does the linear property of definite integrals allow us to do with complex definite integrals?

  Split into simpler parts
• The linearity property holds for both definite and indefinite integrals.

  True
• The constant multiple property simplifies definite integral calculations.


  True
• The sum and difference property allows for easier integration of complex functions.


  True