Integrating Special Functions (DP IB Maths: AA HL)

Revision Note

Author

Paul

Last updated

17 May 2023

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Integrating Trig Functions

How do I integrate sin, cos and sec^2?

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 $\tan x$

$\int \sec^{2} x d x = \tan x + c$

The derivatives of $\sin x, \cos x$ and $\tan x$ are in the formula booklet
- so these antiderivatives can be easily deduced
For the linear function $a x + b$ , where $a$ and $b$ are constants,

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

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

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

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

The formula booklet can be used to find antiderivatives from the derivatives
- Make sure you have the page with the section of standard derivatives open
- Use these backwards to find any antiderivatives you need
- Remember to add 'c', the constant of integration, for any indefinite integrals

Worked example

Find, in the form

F (x) + c

, an expression for each integral

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

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

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

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Previous:5.3.2 Applications of IntegrationNext:5.4.2 Techniques of Integration

Integrating Special Functions (DP IB Maths: AA HL)

Revision Note

How do I integrate sin, cos and sec^2?

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 & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

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 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

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 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities

2.8.1 Solving Inequalities Graphically

2.8.2 Polynomial Inequalities

2.9 Further Functions & Graphs