Integrating Special Functions (DP IB Maths: AI HL)

Revision Note

Author

Paul

Last updated

19 July 2024

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Integrating Trig Functions

How do I integrate sin, cos and 1/cos²?

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

Also, from the derivative of $\tan x$

$\int \frac{1}{\cos^{2} x} d x = \tan x + c$

All three of these standard integrals are in the formula booklet
For the linear function $a x + b$ , where $a$ and $b$ are constants,

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

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

$\int \frac{1}{\cos^{2} (a x + b)} d x = \frac{1}{a} \tan (a x + b) + c$

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

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

Find, in the form

F (x) + c

, an expression for each integral

$\int \cos x d x$
$\int \frac{1}{\cos^{2} (3 x - \frac{π}{3})} d x$

5-4-1-ib-hl-ai-aa-extraaa-ai-we1a-soltn

b) A curve has equation

y = \int 2 \sin (2 x + \frac{π}{6}) d x

The curve passes through the point with coordinates

(\frac{π}{3}, \sqrt{3})

Find an expression for

y

5-4-1-ib-hl-ai-aa-extraaa-we1b-soltn-

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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

When revising, familiarise yourself with the layout of this section of the formula booklet, make sure you know what is and isn't in there and how to find it very quickly

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

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Previous:5.3.3 Applications of IntegrationNext:5.4.2 Techniques of Integration

Integrating Special Functions (DP IB Maths: AI HL)

Revision Note

How do I integrate sin, cos and 1/cos2?

How do I integrate exponentials and 1/x?

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

1. Number & Algebra

1.1 Number Toolkit

1.1.1 Standard Form

1.1.2 Approximation & Estimation

1.1.3 GDC: Solving Equations

1.2 Exponentials & Logs

1.2.1 Exponents

1.2.2 Logarithms

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

1.4 Financial Applications

1.4.1 Compound Interest & Depreciation

1.4.2 Amortisation & Annuities

1.5 Complex Numbers

1.5.1 Intro to Complex Numbers

1.5.2 Modulus & Argument

1.5.3 Introduction to Argand Diagrams

1.6 Further Complex Numbers

1.6.1 Geometry of Complex Numbers

1.6.2 Forms of Complex Numbers

1.6.3 Applications of Complex Numbers

1.7 Matrices

1.7.1 Introduction to Matrices

1.7.2 Operations with Matrices

1.7.3 Determinants & Inverses

1.7.4 Solving Systems of Linear Equations with Matrices

1.8 Eigenvalues & Eigenvectors

1.8.1 Eigenvalues & Eigenvectors

1.8.2 Applications of Matrices

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Further Functions & Graphs

2.2.1 Functions

2.2.2 Graphing Functions

2.2.3 Properties of Graphs

2.3 Modelling with Functions

2.3.1 Linear Models

2.3.2 Quadratic & Cubic Models

2.3.3 Exponential Models

2.3.4 Direct & Inverse Variation

2.3.5 Sinusoidal Models

2.3.6 Strategy for Modelling Functions

2.4 Functions Toolkit

2.4.1 Composite & Inverse 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

2.6 Further Modelling with Functions

2.6.1 Properties of Further Graphs

2.6.2 Natural Logarithmic Models

2.6.3 Logistic Models

2.6.4 Piecewise Models

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 Simple Identities

How do I integrate sin, cos and 1/cos²?