antiderivative of rational function

antiderivative of rational function

The most notable real functions, which can be integrated in a

closed form

, are the

rational functions

:

Theorem

.

The

antiderivative

of a rational function is always expressible in a closed form, which only can comprise, except a rational expression summand, summands of

logarithms

and arcustangents of rational functions.

One can justify the theorem by using the general form of the (unique)

partial fraction decomposition

R

(

x

)

=

H

(

x

)

+

∑

i

=

1

m

(

A

i

⁢

1

x

-

a

i

+

A

i

⁢

2

(

x

-

a

i

)

2

+

…

+

A

i

⁢

μ

i

(

x

-

a

i

)

μ

i

)

+

∑

j

=

1

n

(

B

j

⁢

1

⁢

x

+

C

j

⁢

1

x

2

+

2

⁢

p

j

⁢

x

+

q

j

+

B

j

⁢

2

⁢

x

+

C

j

⁢

2

(

x

2

+

2

⁢

p

j

⁢

x

+

q

j

)

2

+

…

+

B

j

⁢

ν

j

⁢

x

+

C

j

⁢

ν

j

(

x

2

+

2

⁢

p

j

⁢

x

+

q

j

)

ν

j

)

,

of the rational function

R

⁢

(

x

)

; here,

H

⁢

(

x

)

is a

polynomial

, the first

sum expression

is determined by the real zeroes

a

i

of the

denominator

of

R

⁢

(

x

)

, the second sum is determined by the real quadratic

prime factors

x

2

+

2

⁢

p

j

⁢

x

+

q

j

of the denominator (which have no real zeroes).

The addends of the form

A

(

x

-

a

)

r

in the first sum are integrated directly, giving

∫

A

x

-

a

d

x

=

A

ln

|

x

-

a

|

+

constant

(

r

=

1

)

(1)

and

∫

A

(

x

-

a

)

r

d

x

=

-

A

r

-

1

⋅

1

(

x

-

a

)

r

-

1

+

constant

(

r

>

1

)

.

(2)

The remaining

partial fractions

are of the form

B

⁢

x

+

C

(

x

2

+

2

⁢

p

⁢

x

+

q

)

s

where

p

2

<

q

and

s

is a

positive integer

.  Now we may write

x

2

+

2

⁢

p

⁢

x

+

q

=

(

x

+

p

)

2

+

q

-

p

2

=

(

q

-

p

2

)

⁢

[

1

+

(

x

+

p

q

-

p

2

)

2

]

and make the substitution

x

+

p

q

-

p

2

=

t

,

(3)

i.e.

x

=

t

⁢

q

-

p

2

-

p

,  getting

∫

B

⁢

x

+

C

(

x

2

+

2

⁢

p

⁢

x

+

q

)

s

⁢

𝑑

x

=

∫

E

⁢

t

+

F

(

1

+

t

2

)

s

⁢

𝑑

t

=

E

⁢

∫

t

⁢

d

⁢

t

(

1

+

t

2

)

s

+

F

⁢

∫

d

⁢

t

(

1

+

t

2

)

s

(4)

where

E

and

F

are certain constants.  In the case

s

=

1

we have

∫

t

⁢

d

⁢

t

1

+

t

2

=

1

2

⁢

ln

⁡

(

1

+

t

2

)

+

constant

(5)

and in the case

s

>

1

∫

t

⁢

d

⁢

t

(

1

+

t

2

)

s

=

-

1

2

⁢

(

s

-

1

)

⋅

1

(

1

+

t

2

)

s

-

1

+

constant

.

(6)

The latter addend of the right hand side of (4) is for

s

=

1

got from

∫

d

⁢

t

1

+

t

2

=

arctan

⁡

t

+

constant

(7)

and for the cases

s

>

1

on may first write

∫

d

⁢

t

(

1

+

t

2

)

s

=

∫

(

1

+

t

2

)

-

t

2

(

1

+

t

2

)

s

⁢

𝑑

t

=

∫

d

⁢

t

(

1

+

t

2

)

s

-

1

-

∫

t

⋅

t

⁢

d

⁢

t

(

1

+

t

2

)

s

.

Using

integration by parts

in the last

integral

, this equation can be converted into the

reduction formula

∫

d

⁢

t

(

1

+

t

2

)

s

=

1

2

⁢

s

-

2

⋅

t

(

1

+

t

2

)

s

-

1

+

2

⁢

n

-

3

2

⁢

n

-

2

⁢

∫

d

⁢

t

(

1

+

t

2

)

s

-

1

.

(8)

The assertion of the theorem follows from (1), …, (8).

Example.

∫

d

⁢

x

1

+

x

4

=

1

4

⁢

2

⁢

ln

⁡

1

+

x

⁢

2

+

x

2

1

-

x

⁢

2

+

x

2

+

1

2

⁢

2

⁢

arctan

⁡

x

⁢

2

1

-

x

2

+

C

Title

antiderivative of rational function

Canonical

name

AntiderivativeOfRationalFunction

Date of creation

2013-03-22 19:21:38

Last modified on

2013-03-22 19:21:38

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

19

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A36

Synonym

integration of rational functions

Related topic

IntegrationTechniques

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