Prog 15 Integration MEF

INTEGRATION 1

STROUD

Worked examples and exercises are in the text

Integration 1
Introduction

Functions of a linear function of x

Integrals of the form f ( x ) / f ( x ) and f ( x ). f ( x )

Integration of products – integration by parts
Integration by partial fractions
Integration of trigonometric functions

STROUD

Worked examples and exercises are in the text

Integration 1
Introduction

Functions of a linear function of x

Integrals of the form f ( x ) / f ( x ) and f ( x ). f ( x )

Integration of products – integration by parts
Integration by partial fractions
Integration of trigonometric functions

STROUD

Worked examples and exercises are in the text

Integration 1
Introduction

Integration is the reverse process of differentiation. For example:
d 3

( x )  3 x2 and
dx

2
3
x
dx
x
3

C


where C is called the constant of integration.

STROUD

Worked examples and exercises are in the text

Integration 1

Introduction
Standard integrals

What follows is a list of basic derivatives and associated basic integrals:
d n
( x )  nxn 1
dx
1
d
(ln x) 
dx
x
d x
(e )  e x
dx
d kx
(e )  ke kx
dx

STROUD

xn 1
 x dx  n  1  C
1
 x dx  ln x  C
n

x
x
e
dx
e

C


e kx
 e dx  k  C
kx

Worked examples and exercises are in the text

Integration 1
Introduction
Standard integrals

d x
(a )  a x ln a
dx
d
(cos x)   sin x
dx
d
(sin x)  cos x
dx
d
(tan x)  sec 2 x
dx

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ax
 a dx  ln a  C
x

 sin xdx   cos x  C
 cos xdx  sin x  C

2
sec
 xdx  tan x  C

Worked examples and exercises are in the text

Integration 1
Introduction
Standard integrals

d
(cosh x)  sinh x

dx
d
(sinh x)  cosh x
dx
d
1
(sin 1 x) 
dx
1  x2

d
1
(cos 1 x) 
dx
1  x2

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 sinh xdx  cosh x  C




 cosh xdx  sinh x  C
1

1  x2
1

1  x2

dx  sin 1 x  C

dx  cos 1 x  C

Worked examples and exercises are in the text

Integration 1
Introduction
Standard integrals

d
1
(tan 1 x) 
dx
1  x2
d
1
(sinh 1 x) 
dx
x2  1

d
(cosh 1 x) 
dx

1

x2  1
d

1
(tanh 1 x) 
dx
1  x2

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1
1
 1  x2 dx  tan x  C
1
1
 x2  1 dx  sinh x  C
1
1
dx
xC
cosh

 x2  1

1
1
dx
xC
tanh

 1  x2

Worked examples and exercises are in the text

Integration 1
Introduction

Functions of a linear function of x

Integrals of the form f ( x ) / f ( x ) and f ( x ). f ( x )

Integration of products – integration by parts
Integration by partial fractions
Integration of trigonometric functions

STROUD

Worked examples and exercises are in the text

Integration 1
Functions of a linear function of x

If:



then:

For example:

 f ( x)dx  F ( x)  C
f (ax  b)dx 

x7
 x dx  7  C so that
6

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F (ax  b)
C
a

(5 x  4)7
 (5x  4) dx  7  5  C
6

Worked examples and exercises are in the text

Integration 1
Introduction

Functions of a linear function of x

Integrals of the form f ( x ) / f ( x ) and f ( x ). f ( x )

Integration of products – integration by parts
Integration by partial fractions
Integration of trigonometric functions

STROUD

Worked examples and exercises are in the text

Integration 1
Integrals of the form f ( x ) / f ( x ) and f ( x ). f ( x )

(a)



f ( x)
1
dx  
df ( x)  ln f ( x)  C
f ( x)
f ( x)

For example:

(b)

d ( x2  3x  5)
2x  3
2
 x2  3x  5 dx   x2  3x  5  ln  x  3x  5  C

 f ( x) f ( x)dx  

For example:

STROUD

 f ( x)
f ( x)df ( x) 

2

2

C

tan 2 x
 tan xsec xdx   tan xd (tan x)  2  C
2

Worked examples and exercises are in the text

Integration 1
Introduction

Functions of a linear function of x

Integrals of the form f ( x ) / f ( x ) and f ( x ). f ( x )

Integration of products – integration by parts
Integration by partial fractions
Integration of trigonometric functions

STROUD

Worked examples and exercises are in the text

Integration 1
Integration of products – integration by parts

The parts formula is:

 u( x)dv( x)  u( x)v( x)   v( x)du( x)

For example:

x
xe
 dx  u ( x)dv( x)

 u ( x)v( x)   v( x)du ( x) where u ( x)  x so du ( x)  dx
 x.e x   e xdx

dv( x)  e xdx so v( x)  e x

 xe x  e x  C

STROUD

Worked examples and exercises are in the text

Integration 1
Introduction

Functions of a linear function of x

Integrals of the form f ( x ) / f ( x ) and f ( x ). f ( x )

Integration of products – integration by parts
Integration by partial fractions
Integration of trigonometric functions

STROUD

Worked examples and exercises are in the text

Integration 1
Integration by partial fractions

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:
x 1
2 
 3
dx


 x2  3x  2   x  2 x  1  dx
3
2
dx  
dx

x2
x 1
 3ln( x  2)  2ln( x  1)  C

STROUD

Worked examples and exercises are in the text

Integration 1
Introduction

Functions of a linear function of x

Integrals of the form f ( x ) / f ( x ) and f ( x ). f ( x )

Integration of products – integration by parts
Integration by partial fractions
Integration of trigonometric functions

STROUD

Worked examples and exercises are in the text

Integration 1
Integration of trigonometric functions

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

For example:

STROUD

2
sin
 xdx  

1
1  cos 2 x dx
2
1
1
  dx   cos 2 xdx
2
2
x sin 2 x
 
C
2
4

Worked examples and exercises are in the text

Integration 1
Learning outcomes

Integrate standard expressions using a table of standard forms
Integrate functions of a linear form
Evaluate integrals with integrands of the form f ( x ) / f ( x ) and f ( x ). f ( x )
Integrate by parts
Integrate by partial fractions
Integrate trigonometric functions

STROUD

Worked examples and exercises are in the text