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DP IB Maths: AA SL

Revision Notes

Home / IB / Maths: AA SL / DP / Revision Notes / 5. Calculus / 5.4 Further Integration / 5.4.1 Integrating Special Functions


5.4.1 Integrating Special Functions


Integrating Trig Functions

How do I integrate sin and cos?

  • The antiderivatives for sine and cosine are

bold space bold integral bold sin bold space bold italic x bold space bold d bold italic x bold equals bold minus bold cos bold space bold italic x bold plus bold italic c

bold space bold integral bold cos bold space bold italic x bold space bold d bold italic x bold equals bold sin bold space bold italic x bold plus bold italic c

wherebold space bold italic c is the constant of integration

    • These are given in the formula booklet
  • For the linear functionbold space bold italic a bold italic x bold plus bold italic b, wherespace bold italic a andspace bold italic b are constants,

bold space bold integral bold sin bold space bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis bold space bold d bold italic x bold equals bold minus bold 1 over bold italic a bold cos bold space bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis bold plus bold italic c

bold space bold integral bold cos bold space bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis bold space bold d bold italic x bold equals bold 1 over bold italic a bold sin bold space bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis bold plus bold italic c

  • For calculus with trigonometric functions angles must be measured in radians
    • Ensure you know how to change the angle mode on your GDC

Worked Example

a)
Find, in the formspace straight F left parenthesis x right parenthesis plus c, an expression for each integral
i)
space integral cos space x space straight d x
ii)
space integral 3 sin stretchy left parenthesis 2 x plus straight pi over 3 stretchy right parenthesis space straight d x
iii)
space integral left parenthesis 2 sin left parenthesis 4 x right parenthesis minus 3 cos left parenthesis 2 x right parenthesis right parenthesis space straight d x

 5-4-1-ib-sl-aa-only-we1-soltn-a

b)
The graph ofspace y equals straight F left parenthesis x right parenthesis plus c for question (a) part (ii) passes through the point with coordinatesspace stretchy left parenthesis straight pi over 3 comma space 5 over 2 stretchy right parenthesis.
Find the value ofspace c.

5-4-1-ib-sl-aa-only-we1-soltn-b

Integrating e^x & 1/x

How do I integrate exponentials and logarithms?

  • The antiderivatives forbold space bold e to the power of bold italic x andspace bold ln bold space bold italic x are

bold space bold integral bold space bold e to the power of bold x bold space bold d bold italic x bold equals bold space bold e to the power of bold x bold plus bold italic c

Error converting from MathML to accessible text.

where bold italic c is the constant of integration

    • These are given in the formula booklet
  • For the linear functionbold space bold left parenthesis bold italic a bold italic x bold plus bold italic b bold right parenthesis, wherespace bold italic a andspace bold italic b are constants,

 bold space bold integral bold e to the power of bold italic a bold italic x bold italic plus bold italic b end exponent bold space bold d bold italic x bold equals bold 1 over bold italic a bold e to the power of bold italic a bold italic x bold italic plus bold italic b end exponent bold plus bold italic c

  • It follows from the last result that

 

    • which can be deduced using Reverse Chain Rule
  • With ln, it can be useful to write the constant of integration,space c, as a logarithm
    • using the laws of logarithms, the answer can be written as a single term
    • wherespace k is a constant
    • This is similar to the special case of differentiatingspace ln space left parenthesis a x plus b right parenthesis whenspace b equals 0

Exam Tip

  • Make sure you have a copy of the formula booklet during revision but don't try to remember everything in the formula booklet
    • However, do be familiar with the layout of the formula booklet – you’ll be able to locate quickly whatever you are after, and you do not want to be searching every line of every page!
    • For formulae you think you have remembered, use the booklet to double-check

Worked Example

A curve has the gradient functionspace straight f apostrophe left parenthesis x right parenthesis equals fraction numerator 3 over denominator 3 x plus 2 end fraction plus e to the power of 4 minus x end exponent.


Given the exact value ofspace straight f left parenthesis 1 right parenthesis isspace ln space 10 minus e cubed find an expression forspace straight f left parenthesis x right parenthesis.

5-4-1-ib-sl-aa-only-we2-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
                                                                    • 3.1.3 Arcs & Sectors
                                                                    • 3.2 Geometry of 3D Shapes
                                                                      • 3.2.1 3D Coordinate Geometry
                                                                        • 3.2.2 Volume & Surface Area
                                                                        • 3.3 Trigonometry
                                                                          • 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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