Integrating Special Functions | DP IB Maths: AA SL Revision Notes 2021

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Integrating Trig Functions

How do I integrate sin and cos?

The antiderivatives for sine and cosine are

$\int \sin x d x = - \cos x + c$

$\int \cos x d x = \sin x + c$

where $c$ is the constant of integration

- These are given in the formula booklet
For the linear function $a x + b$ , where $a$ and $b$ are constants,

$\int \sin (a x + b) d x = - \frac{1}{a} \cos (a x + b) + c$

$\int \cos (a x + b) d x = \frac{1}{a} \sin (a x + b) + c$

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

Find, in the form

F (x) + c

, an expression for each integral

\int \cos x d x

ii)

\int 3 \sin (2 x + \frac{π}{3}) d x

iii)

\int (2 \sin (4 x) - 3 \cos (2 x)) d x

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

The graph of

y = F (x) + c

for question (a) part (ii) passes through the point with coordinates

(\frac{π}{3}, \frac{5}{2})

Find the value of

c

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

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Integrating e^x & 1/x

How do I integrate exponentials and 1/x?

The antiderivatives involving $e^{x}$ and $\ln x$ are

$\int e^{x} d x = e^{x} + c$

$\int \frac{1}{x} d x = \ln | x | + c$

where $c$ is the constant of integration

- These are given in the formula booklet
For the linear function $(a x + b)$ , where $a$ and $b$ are constants,

$\int e^{a x + b} d x = \frac{1}{a} e^{a x + b} + c$

$\int \frac{1}{a x + b} d x = \frac{1}{a} \ln | a x + b | + c$

It follows from the last result that

$\int \frac{a}{a x + b} d x = \ln | a x + b | + c$

- which can be deduced using Reverse Chain Rule
With ln, it can be useful to write the constant of integration, $c$ , as a logarithm
- using the laws of logarithms, the answer can be written as a single term
- $\int \frac{1}{x} d x = \ln | x | + \ln k = \ln k | x |$ where $k$ is a constant
- This is similar to the special case of differentiating $\ln (a x + b)$ when $b = 0$

Examiner 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 quickly locate whatever you are after
  - 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 function $f' (x) = \frac{3}{3 x + 2} + e^{4 - x}$ .

Given the exact value of $f (1)$ is $\ln 10 - e^{3}$ find an expression for $f (x)$ .

NA5HYQ75_5-4-1-ib-sl-aa-only-we2-soltn

Integrating Special Functions (DP IB Maths: AA SL)

Revision Note

How do I integrate sin and cos?

How do I integrate exponentials and 1/x?

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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 Trigonometry

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