1.8 Integration

• Integration is the reverse process of differentiation
• What is the mathematical expression for the integral of f(x)?

  $\int f(x) \, dx$
• The constant of integration arises because the derivative of a constant is zero.
• The fundamental theorem of calculus consists of two parts
• What is the formula for Part 1 of the fundamental theorem of calculus?

  $\int_{a}^{b} f(x) \, dx =$$F(b) - F(a)$
• Part 2 of the fundamental theorem of calculus states that the derivative of $F(x) =$$\int_{a}^{x} f(t) \, dt$ is f(x)
• The constant of integration is needed in definite integrals.

  False
• The power rule for integration states that $\int x^{n} \, dx =$$\frac{x^{n + 1}}{n + 1} +$$C$, where $C$ is the constant of integration
• What is the integral of $e^{x}$?


  e^{x} + C</latex>
• Match the trigonometric function with its integral:

  \sin x ↔️ - \cos x + C

  \cos x ↔️ \sin x + C
• What is the mathematical formula for integration?

  $\int f(x) \, dx =$$F(x) +$$C$
• Integration is the reverse process of differentiation
• The integral of a function represents the area under its curve.
• What is the integral of $f(x) =$$2x$?


  $x^{2} +$$C$
• Steps to apply the fundamental theorem of calculus (Part 1)

  1️⃣ Find the antiderivative $F(x)$

  2️⃣ Evaluate $F(b)$ and $F(a)$

  3️⃣ Calculate $F(b) - F(a)$
• The constant of integration is needed in definite integrals.

  False
• What is the power rule for integration?

  $\int x^{n} \, dx =$$\frac{x^{n + 1}}{n + 1} +$$C$
• What is the integral of $\sin x$?


  $- \cos x +$$C$
• The derivative of any constant is zero.
• Steps to find the area under a curve using integration

  1️⃣ Determine the limits of integration

  2️⃣ Set up the definite integral

  3️⃣ Find the antiderivative

  4️⃣ Evaluate the antiderivative at the limits
• What is the area under the curve $f(x) =$$x^{2}$ from x = 1</latex> to $x =$$3$?


  $\frac{26}{3}$