STROUD

Worked examples and exercises are in the text

PROGRAMME 15

INTEGRATION 1

STROUD

Worked examples and exercises are in the text

Introduction

Functions of a linear function of x

Integrals of the form and

Integration of products – integration by parts

Integration by partial fractions

Integration of trigonometric functions

Programme 15: Integration 1

( ) / ( )f x f x ( ). ( )f x f x

STROUD

Worked examples and exercises are in the text

Introduction

Functions of a linear function of x

Integrals of the form and

Integration of products – integration by parts

Integration by partial fractions

Integration of trigonometric functions

Programme 15: Integration 1

( ) / ( )f x f x ( ). ( )f x f x

STROUD

Worked examples and exercises are in the text

Introduction

Programme 15: Integration 1

Integration is the reverse process of differentiation. For example:

where C is called the constant of integration.

3 2 2 3( ) 3 and 3 d

x x x dx x Cdx

STROUD

Worked examples and exercises are in the text

Introduction

Standard integrals

Programme 15: Integration 1

What follows is a list of basic derivatives and associated basic integrals:

11( )

11 1

(ln ) ln

( )

( )

nn n n

x x x x

kxkx kx kx

d xx nx x dx C

dx nd

x dx x Cdx x xd

e e e dx e Cdx

d ee ke e dx C

dx k

STROUD

Worked examples and exercises are in the text

Introduction

Standard integrals

Programme 15: Integration 1

2 2

( ) lnln

(cos ) sin sin cos

(sin ) cos cos sin

(tan ) sec sec tan

xx x xd a

a a a a dx Cdx a

dx x xdx x C

dxd

x x xdx x Cdxd

x x xdx x Cdx

STROUD

Worked examples and exercises are in the text

Introduction

Standard integrals

Programme 15: Integration 1

1 1

2 2

1 1

2 2

(cosh ) sinh sinh cosh

(sinh ) cosh cosh sinh

1 1(sin ) sin

1 11 1

(cos ) cos1 1

dx x xdx x C

dxd

x x xdx x Cdxd

x dx x Cdx x xd

x dx x Cdx x x

STROUD

Worked examples and exercises are in the text

Introduction

Standard integrals

Programme 15: Integration 1

1 12 2

1 1

2 2

1 1

2 2

1 12 2

1 1(tan ) tan

1 11 1

(sinh ) sinh1 1

1 1(cosh ) cosh

1 11 1

(tanh ) tanh1 1

dx dx x C

dx x xd

x dx x Cdx x xd

x dx x Cdx x x

dx dx x C

dx x x

STROUD

Worked examples and exercises are in the text

Introduction

Functions of a linear function of x

Integrals of the form and

Integration of products – integration by parts

Integration by partial fractions

Integration of trigonometric functions

Programme 15: Integration 1

( ) / ( )f x f x ( ). ( )f x f x

STROUD

Worked examples and exercises are in the text

Functions of a linear function of x

Programme 15: Integration 1

If:

then:

For example:

( ) ( ) f x dx F x C

( )( )

F ax bf ax b dx C

a

7 76 6 (5 4)

so that (5 4)7 7 5

x x

x dx C x dx C

STROUD

Worked examples and exercises are in the text

Introduction

Functions of a linear function of x

Integrals of the form and

Integration of products – integration by parts

Integration by partial fractions

Integration of trigonometric functions

Programme 15: Integration 1

( ) / ( )f x f x ( ). ( )f x f x

STROUD

Worked examples and exercises are in the text

Integrals of the form and

Programme 15: Integration 1

( ) / ( )f x f x ( ). ( )f x f x

(a)

For example:

(b)

For example:

( ) 1( ) ln ( )

( ) ( )

f xdx df x f x C

f x f x

2

22 2

2 3 ( 3 5)ln 3 5

3 5 3 5

x d x x

dx x x Cx x x x

2( )

( ) ( ) ( ) ( )2

f x

f x f x dx f x df x C

22 tan

tan sec tan (tan )2

x

x xdx xd x C

STROUD

Worked examples and exercises are in the text

Introduction

Functions of a linear function of x

Integrals of the form and

Integration of products – integration by parts

Integration by partial fractions

Integration of trigonometric functions

Programme 15: Integration 1

( ) / ( )f x f x ( ). ( )f x f x

STROUD

Worked examples and exercises are in the text

Integration of products – integration by parts

Programme 15: Integration 1

The parts formula is:

For example:

( ) ( ) ( ) ( ) ( ) ( ) u x dv x u x v x v x du x

( ) ( )

( ) ( ) ( ) ( ) where ( ) so ( )

( ) so ( )

.

x

x x

x x

x x

xe dx u x dv x

u x v x v x du x u x x du x dx

dv x e dx v x e

x e e dx

xe e C

STROUD

Worked examples and exercises are in the text

Introduction

Functions of a linear function of x

Integrals of the form and

Integration of products – integration by parts

Integration by partial fractions

Integration of trigonometric functions

Programme 15: Integration 1

( ) / ( )f x f x ( ). ( )f x f x

STROUD

Worked examples and exercises are in the text

Integration by partial fractions

Programme 15: Integration 1

If the integrand is an algebraic fraction that can be separated into its partial fractions then each individual partial fraction can be integrated separately.

For example:

2

1 3 2

3 2 2 1

3 2

2 13ln( 2) 2ln( 1)

xdx dx

x x x x

dx dxx x

x x C

STROUD

Worked examples and exercises are in the text

Introduction

Functions of a linear function of x

Integrals of the form and

Integration of products – integration by parts

Integration by partial fractions

Integration of trigonometric functions

Programme 15: Integration 1

( ) / ( )f x f x ( ). ( )f x f x

STROUD

Worked examples and exercises are in the text

Integration of trigonometric functions

Programme 15: Integration 1

Many integrals with trigonometric integrands can be evaluated after applying trigonometric identities.

For example:

2 1sin 1 cos2

21 1

cos22 2

sin 2

2 4

xdx x dx

dx xdx

x xC

STROUD

Worked examples and exercises are in the text

Learning outcomes

Integrate standard expressions using a table of standard forms

Integrate functions of a linear form

Evaluate integrals with integrands of the form and

Integrate by parts

Integrate by partial fractions

Integrate trigonometric functions

Programme 15: Integration 1

( ) / ( )f x f x ( ). ( )f x f x