MTH30105/MTH30605 Mathematics Topic 4 – Integration

Prepared by: Tee Lee Hong
Presented by: Afiq Arif Aminuddin Jafry

Outline

  • What is integration
  • Basic rules of integration
  • Integration for power function
  • Integration for rational function

What is Integration?

  • Differential Calculus: Cuts a function into small pieces to analyze how it changes.
  • Integral Calculus: Joins (integrates) those small pieces to determine the total magnitude or accumulation.

Introduction to Integration

  • Integration is instrumental in calculating:

    • Areas under curves
    • Volumes of solids
    • Central points of functions

  • Example: Finding the area under the curve of a function by calculating area at various points.

Area Calculation using Integration

  • To find the area under a function:

    • Method 1: Calculate the area at discrete points and sum them up (using slices of width riangle x) leads to inaccuracies.

    • Method 2: As riangle x becomes very small and the number of slices increases, the approximation accuracy improves.

    • As slices approach zero in width, the result hones in on the true area under the curve. This is denoted as:

    • dx indicates that the width of slices is approaching zero.

Integration Process

  • The integration process can be viewed as the reverse of differentiation.
  • For instance, to integrate the function 2x with respect to x:
    • The integral is expressed as:
      \int 2x \, dx = x^2 + C
  • Constant of Integration (C): Added to the result to account for all functions whose derivative yields the same result. Examples include:
    • \frac{d}{dx}(x^2 + 4) = 2x and \frac{d}{dx}(x^2 + 99) = 2x
    • Since the derivative of a constant is zero, the general form of the integral is 2x + C where C can be any constant.

Learning Outcomes

  • Ability to find the antiderivative of a function given its known derivative.
  • Application of basic rules of integration in solving mathematical problems.

Antiderivatives from Known Derivatives

  • Consider a function y = x^2 - 3x. The derivative is:
    \frac{dy}{dx} = 2x - 3
  • The antiderivative (indefinite integral) is given by:
    \int (2x - 3) \, dx = x^2 - 3x + C
  • Thus, we can conclude:
    • \int f'(x) \, dx = f(x) + C where C is a constant.

Examples of Antiderivatives

  • Example 1: Find:
    \int 2x \, dx

    • Solution: \int 2x \, dx = x^2 + C

  • Example 2: Integrate:
    \int x^2 \, dx

    • Solution: \frac{x^3}{3} + C

Basic Rules of Integration

  • Basic Formulas:
    • \int 1 \, dx = x + C
    • \int a \, dx = ax + C where a is a constant.
    • \int x^n \, dx = \frac{x^{n+1}}{n+1} + C for n \neq -1.
    • \int a f(x) \, dx = a \int f(x) \, dx
    • \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx

Examples of Basic Integration Rules

  • Integrate the following:
    • a. \int 4x^2 \, dx
    • Solution: \frac{4x^3}{3} + C
    • b. \int (t^3 + 4t^2) \, dt
    • Solution: \frac{t^4}{4} + \frac{4t^3}{3} + C
    • c. \int 2x \, dx
    • Solution: x^2 + C
    • d. \int (2w + 1) \, dw
    • Solution: w^2 + w + C

Types of Functions in Integration

  • Power function
  • Rational function
  • Exponential function
  • Trigonometric function
  • Others including mixed types or higher-degree polynomial expressions.

Power Function Integration

  • Basic Integration Techniques:
    • Apply fundamental rules of integration.
    • Use integration by substitution for more complex functions.

Power Function Integration by Substitution - Example 1

  • Integrate: \int \sqrt{(2x + 1)^2} \, dx
    • Expand to obtain:
    • \int (4x^2 + 4x + 1) \, dx
    • Proceed with integration observing power rules:
    • = \frac{4}{3}x^3 + 2x^2 + x + C

Substitution Integration Example Process

  • Let: u = 2x + 1 \Rightarrow du = 2dx ightarrow dx = \frac{1}{2} du
    • Update the integral as:
      \int \frac{1}{2} u^2 \, du = \frac{1}{2} \frac{u^3}{3} + C
    • Resubstituting yields the antiderivative back to x.

Rational Function Integration

  • Key Function Types:
    • Power Function
    • Rational Function
    • Exponential and Trigonometric Functions
    • Composite Functions (e.g., \frac{x^2}{2x^2 - x - 1})

Integration of Rational Functions

  • Can often be solved via:
    • Integration by substitution.
    • Partial fraction decomposition.

Rational Function Example - Integration by Substitution

  • For instance, to integrate: \int (4x + 5)^3 \, dx
    • Taking derivative and applying substitution leads to simplifying integrals and finding a solution with constants included.

Integration by Partial Fractions Key Steps

  • To integrate a function like:
    \int \frac{3x}{2x^2 - x - 1} \, dx

    • Begin by performing partial fraction decomposition:

    • Set:
      \frac{3x}{2x^2 - x - 1} = \frac{A}{2x + 1} + \frac{B}{x - 1}

    • For determination, up the values to isolate and solve for constants A and B, and continue toward integrating each term.

Conclusion on Partial Fractions in Integration

  • Integration by partial fractions is vital for rational functions, especially when the denominator has linear and possibly repeated factors.
  • Through methodical substitution and breakdown, integration becomes manageable and yields final results for complex functions of rational forms.