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DP IB Maths: AA SL

Revision Notes

Home / IB / Maths: AA SL / DP / Revision Notes / 5. Calculus / 5.3 Integration / 5.3.1 Introduction to Integration


5.3.1 Introduction to Integration


Introduction to Integration

What is integration?

  • Integration is the opposite to differentiation
    • Integration is referred to as antidifferentiation
    • The result of integration is referred to as the antiderivative
  • Integration is the process of finding the expression of a function (antiderivative) from an expression of the derivative (gradient function)

What is the notation for integration?

  • An integral is normally written in the form

integral f left parenthesis x right parenthesis space straight d x 

    • the large operator integral means “integrate”
    • “straight d x” indicates which variable to integrate with respect to
    • space f left parenthesis x right parenthesis is the function to be integrated (sometimes called the integrand)
  • The antiderivative is sometimes denoted by straight F left parenthesis x right parenthesis
    • there’s then no need to keep writing the whole integral; refer to it as straight F left parenthesis x right parenthesis
  • straight F left parenthesis x right parenthesis may also be called the indefinite integral ofspace f left parenthesis x right parenthesis

What is the constant of integration? 

  • Recall one of the special cases from Differentiating Powers of x
    • Ifspace f left parenthesis x right parenthesis equals a thenspace f apostrophe left parenthesis x right parenthesis equals 0
  • This means that integrating 0 will produce a constant term in the antiderivative
    • a zero term wouldn’t be written as part of a function
    • every function, when integrated, potentially has a constant term
  • This is called the constant of integration and is usually denoted by the letter c
    • it is often referred to as “plus c”
  • Without more information it is impossible to deduce the value of this constant
    • there are endless antiderivatives, straight F left parenthesis x right parenthesis, for a functionspace f left parenthesis x right parenthesis

Integrating Powers of x

How do I integrate powers of x?

  • Powers ofspace x are integrated according to the following formulae:
    • Ifspace f left parenthesis x right parenthesis equals x to the power of n thenspace integral f left parenthesis x right parenthesis space straight d x equals fraction numerator x to the power of n plus 1 end exponent over denominator n plus 1 end fraction plus c wherespace n element of straight rational numbers comma space n not equal to negative 1 andspace c is the constant of integration

    • This is given in the formula booklet
  • If the power ofspace x is multiplied by a constant then the integral is also multiplied by that constant
    • Ifspace f left parenthesis x right parenthesis equals a x to the power of n thenspace integral f left parenthesis x right parenthesis space straight d x equals fraction numerator a x to the power of n plus 1 end exponent over denominator n plus 1 end fraction plus c wherespace n element of straight rational numbers comma space n not equal to negative 1 andspace a is a constant andspace c is the constant of integration
  • fraction numerator straight d y over denominator straight d x end fraction notation can still be used with integration
  • Note that the formulae above do not apply whenspace x equals negative 1 as this would lead to division by zero
  • Remember the special case:
    • space integral a space straight d x equals a x plus c
      • e.g. space integral 4 space straight d x equals 4 x plus c 
    • This allows constant terms to be integrated
  • Functions involving roots will need to be rewritten as fractional powers ofspace x first
    • eg. Ifspace f left parenthesis x right parenthesis equals 5 cube root of x then rewrite asspace f left parenthesis x right parenthesis equals 5 x to the power of 1 third end exponent and integrate
  • Functions involving fractions with denominators in terms ofbold space bold italic x will need to be rewritten as negative powers ofspace x first
    • e.g.  Ifspace f left parenthesis x right parenthesis equals 4 over x squared plus x squared then rewrite asspace f left parenthesis x right parenthesis equals 4 x to the power of negative 2 end exponent plus x squared and integrate    

  • The formulae for integrating powers ofspace x apply to all rational numbers so it is possible to integrate any expression that is a sum or difference of powers ofspace x
    • e.g.  Ifspace size 16px f size 16px left parenthesis size 16px x size 16px right parenthesis size 16px equals size 16px 8 size 16px x to the power of size 16px 3 size 16px minus size 16px 2 size 16px x size 16px plus size 16px 4 then
               
  • Products and quotients cannot be integrated this way so would need expanding/simplifying first
    • e.g.  Ifspace f begin mathsize 16px style stretchy left parenthesis x stretchy right parenthesis end style size 16px equals size 16px 8 size 16px x to the power of size 16px 2 size 16px left parenthesis size 16px 2 size 16px x size 16px minus size 16px 3 size 16px right parenthesis then

What might I be asked to do once I’ve found the anti-derivative (integrated)?

  • With more information the constant of integration,space c, can be found
  • The area under a curve can be found using integration

Exam Tip

  • You can speed up the process of integration in the exam by committing the pattern of basic integration to memory
    • In general you can think of it as 'raising the power by one and dividing by the new power'
    • Practice this lots before your exam so that it comes quickly and naturally when doing more complicated integration questions

Worked Example

Given that

find an expression forspace y in terms ofspace x.

5-3-1-ib-sl-aa-version-we1-soltn



  • 1. Number & Algebra
    • 1.1 Number Toolkit
      • 1.1.1 Standard Form
        • 1.1.2 Laws of Indices
        • 1.2 Exponentials & Logs
          • 1.2.1 Introduction to Logarithms
            • 1.2.2 Laws of Logarithms
              • 1.2.3 Solving Exponential Equations
              • 1.3 Sequences & Series
                • 1.3.1 Language of Sequences & Series
                  • 1.3.2 Arithmetic Sequences & Series
                    • 1.3.3 Geometric Sequences & Series
                      • 1.3.4 Applications of Sequences & Series
                        • 1.3.5 Compound Interest & Depreciation
                        • 1.4 Proof & Reasoning
                          • 1.4.1 Proof
                          • 1.5 Binomial Theorem
                            • 1.5.1 Binomial Theorem
                          • 2. Functions
                            • 2.1 Linear Functions & Graphs
                              • 2.1.1 Equations of a Straight Line
                              • 2.2 Quadratic Functions & Graphs
                                • 2.2.1 Quadratic Functions
                                  • 2.2.2 Factorising & Completing the Square
                                    • 2.2.3 Solving Quadratics
                                      • 2.2.4 Quadratic Inequalities
                                        • 2.2.5 Discriminants
                                        • 2.3 Functions Toolkit
                                          • 2.3.1 Language of Functions
                                            • 2.3.2 Composite & Inverse Functions
                                              • 2.3.3 Graphing Functions
                                              • 2.4 Further Functions & Graphs
                                                • 2.4.1 Reciprocal & Rational Functions
                                                  • 2.4.2 Exponential & Logarithmic Functions
                                                    • 2.4.3 Solving Equations
                                                      • 2.4.4 Modelling with Functions
                                                      • 2.5 Transformations of Graphs
                                                        • 2.5.1 Translations of Graphs
                                                          • 2.5.2 Reflections of Graphs
                                                            • 2.5.3 Stretches of Graphs
                                                              • 2.5.4 Composite Transformations of Graphs
                                                            • 3. Geometry & Trigonometry
                                                              • 3.1 Geometry Toolkit
                                                                • 3.1.1 Coordinate Geometry
                                                                  • 3.1.2 Radian Measure
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                                                                          • 3.3.1 Pythagoras & Right-Angled Triganometry
                                                                            • 3.3.2 Non Right-Angled Trigonometry
                                                                              • 3.3.3 Applications of Trigonometry & Pythagoras
                                                                              • 3.4 Further Trigonometry
                                                                                • 3.4.1 The Unit Circle
                                                                                  • 3.4.2 Exact Values
                                                                                  • 3.5 Trigonometric Functions & Graphs
                                                                                    • 3.5.1 Graphs of Trigonometric Functions
                                                                                      • 3.5.2 Transformations of Trigonometric Functions
                                                                                        • 3.5.3 Modelling with Trigonometric Functions
                                                                                        • 3.6 Trigonometric Equations & Identities
                                                                                          • 3.6.1 Simple Identities
                                                                                            • 3.6.2 Double Angle Formulae
                                                                                              • 3.6.3 Relationship Between Trigonometric Ratios
                                                                                                • 3.6.4 Linear Trigonometric Equations
                                                                                                  • 3.6.5 Quadratic Trigonometric Equations
                                                                                                • 4. Statistics & Probability
                                                                                                  • 4.1 Statistics Toolkit
                                                                                                    • 4.1.1 Sampling & Data Collection
                                                                                                      • 4.1.2 Statistical Measures
                                                                                                        • 4.1.3 Frequency Tables
                                                                                                          • 4.1.4 Linear Tranformations of Data
                                                                                                            • 4.1.5 Outliers
                                                                                                              • 4.1.6 Univariate Data
                                                                                                                • 4.1.7 Interpreting Data
                                                                                                                • 4.2 Correlation & Regression
                                                                                                                  • 4.2.1 Bivariate Data
                                                                                                                    • 4.2.2 Correlation & Regression
                                                                                                                    • 4.3 Probability
                                                                                                                      • 4.3.1 Probability & Types of Events
                                                                                                                        • 4.3.2 Conditional Probability
                                                                                                                          • 4.3.3 Sample Space Diagrams
                                                                                                                          • 4.4 Probability Distributions
                                                                                                                            • 4.4.1 Discrete Probability Distributions
                                                                                                                              • 4.4.2 Expected Values
                                                                                                                              • 4.5 Binomial Distribution
                                                                                                                                • 4.5.1 The Binomial Distribution
                                                                                                                                  • 4.5.2 Calculating Binomial Probabilities
                                                                                                                                  • 4.6 Normal Distribution
                                                                                                                                    • 4.6.1 The Normal Distribution
                                                                                                                                      • 4.6.2 Calculations with Normal Distribution
                                                                                                                                        • 4.6.3 Standardisation of Normal Variables
                                                                                                                                      • 5. Calculus
                                                                                                                                        • 5.1 Differentiation
                                                                                                                                          • 5.1.1 Introduction to Differentiation
                                                                                                                                            • 5.1.2 Applications of Differentiation
                                                                                                                                            • 5.2 Further Differentiation
                                                                                                                                              • 5.2.1 Differentiating Special Functions
                                                                                                                                                • 5.2.2 Techniques of Differentiation
                                                                                                                                                  • 5.2.3 Second Order Derivatives
                                                                                                                                                    • 5.2.4 Further Applications of Differentiation
                                                                                                                                                      • 5.2.5 Concavity & Points of Inflection
                                                                                                                                                        • 5.2.6 Derivatives & Graphs
                                                                                                                                                        • 5.3 Integration
                                                                                                                                                          • 5.3.1 Introduction to Integration
                                                                                                                                                            • 5.3.2 Applications of Integration
                                                                                                                                                            • 5.4 Further Integration
                                                                                                                                                              • 5.4.1 Integrating Special Functions
                                                                                                                                                                • 5.4.2 Techniques of Integration
                                                                                                                                                                  • 5.4.3 Definite Integrals
                                                                                                                                                                    • 5.4.4 Further Applications of Integration
                                                                                                                                                                    • 5.5 Optimisation
                                                                                                                                                                      • 5.5.1 Modelling with Differentiation
                                                                                                                                                                      • 5.6 Kinematics
                                                                                                                                                                        • 5.6.1 Kinematics Toolkit
                                                                                                                                                                          • 5.6.2 Calculus for Kinematics
                                                                                                                                                                        Paul Freeman

                                                                                                                                                                        Author: Paul

                                                                                                                                                                        Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.


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