3.9 H: Integration

• Integration reverses the process of differentiation.

  True
• An indefinite integral includes a constant of integration
• Order the steps for integrating a polynomial function.

  1️⃣ Increase the exponent by 1

  2️⃣ Divide by the new exponent

  3️⃣ Add the constant of integration
• The Fundamental Theorem of Calculus is used to evaluate definite integrals.


  True
• What does integration determine given the derivative of a function?

  The original function
• Match the process with its definition.

  Differentiation ↔️ Finding the gradient of a function

  Integration ↔️ Reversing differentiation to find the original function
• A definite integral calculates the area under a curve between two specific points.


  True
• Order the steps for comparing differentiation and integration.

  1️⃣ Define differentiation

  2️⃣ Define integration

  3️⃣ Compare their definitions

  4️⃣ Compare their symbols
• An indefinite integral includes an arbitrary constant
• The integral of $\cos x$ is \sin x
• What is the inverse process of differentiation?

  Integration
• Match the process with its description:

  Differentiation ↔️ Finding the gradient of a function

  Integration ↔️ Reversing differentiation to find the original function
• Does a definite integral require a constant of integration?

  No
• Match the integration rule with its corresponding formula:

  Power Rule ↔️ \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)

  Exponential Rule ↔️ \(\int e^x \, dx = e^x + C\)

  Trigonometric Rule (sin) ↔️ \(\int \sin x \, dx = -\cos x + C\)

  Trigonometric Rule (cos) ↔️ \(\int \cos x \, dx = \sin x + C\)
• The derivative of \( x^3 \) is 3x^2
• The integral of \(\sin(x)\) is \(\cos(x) + C\).

  False
• The derivative of \(\cos(x)\) is \(-\sin(x)\)
• The integral of \(e^x\) is \(e^x + C\).


  True
• Differentiation finds the original function, while integration finds the rate of change.

  False
• Integration determines the original function
• The result of integration is a function plus the constant C
• An indefinite integral includes a constant of integration
• The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)


  True
• The integral of \( \sin x \) is \( -\cos x + C \)


  True
• The integral of \( e^x \) is \( e^x + C \)


  True
• Which type of function can be integrated using basic rules?

  Trigonometric
• What is one application of integration in mathematics?

  Finding areas under curves
• Differentiation finds the rate of change of a function.


  True
• Differentiation involves finding the gradient or rate of change
• A definite integral results in a specific numerical value
• Differentiation involves finding the gradient or rate of change of a function.
• What is the constant of integration denoted by in indefinite integrals?

  C
• What value of \( n \) is excluded in the power rule for integration?

  -1
• The integral of $f'(x)$ is f(x)
• A definite integral represents the area under a curve between two specified points
• An indefinite integral requires adding \( C \) to represent the constant of integration
• Arrange the following features of indefinite and definite integrals in the correct order:

  1️⃣ Definition: General antiderivative with arbitrary constant ||| Definition: Area under curve between two points

  2️⃣ Symbol: \( \int f'(x) \, dx \) ||| Symbol: \( \int_a^b f'(x) \, dx \)

  3️⃣ Result: Function \( f(x) + C \) ||| Result: Value \( f(b) - f(a) \)

  4️⃣ Constant: Requires adding \( C \) ||| Constant: Does not require a constant
• To integrate polynomial functions, we apply the power rule of integration.
• The integral of \( 2x^3 + 5x^2 - x + 4 \) is \( 2 \cdot \frac{x^4}{4} + 5 \cdot \frac{x^3}{3} - \frac{x^2}{2} + 4x + C \), which simplifies to \(\frac{x^4}{2}\)
• The integral of \(\cos(x)\) is \(\sin(x) + C\).

  True