integration of rational function of sine and cosine

integration of rational function of sine and cosine

The integration task

∫

R

⁢

(

sin

⁡

x

,

cos

⁡

x

)

⁢

𝑑

x

,

(1)

where the integrand is a

rational function

of

sin

⁡

x

and

cos

⁡

x

, changes via the

Weierstrass substitution

tan

⁡

x

2

=

t

(2)

to a form having an integrand that is a rational function of

t

.  Namely, since

x

=

2

⁢

arctan

⁡

t

,  we have

d

⁢

x

=

2

⋅

1

1

+

t

2

⁢

d

⁢

t

,

(3)

and we can substitute

sin

⁡

x

=

2

⁢

t

1

+

t

2

,

cos

⁡

x

=

1

-

t

2

1

+

t

2

,

(4)

getting

∫

R

⁢

(

sin

⁡

x

,

cos

⁡

x

)

⁢

𝑑

x

=

2

⁢

∫

R

⁢

(

2

⁢

t

1

+

t

2

,

1

-

t

2

1

+

t

2

)

⁢

d

⁢

t

1

+

t

2

.

Proof

of the formulae (4):  Using the

double angle formulas

of sine and cosine and then dividing the numerators and the denominators by

cos

2

⁡

x

2

we obtain

sin

⁡

x

=

2

⁢

sin

⁡

x

2

⁢

cos

⁡

x

2

sin

2

⁡

x

2

+

cos

2

⁡

x

2

=

2

⁢

tan

⁡

x

2

1

+

tan

2

⁡

x

2

=

2

⁢

t

1

+

t

2

,

cos

⁡

x

=

cos

2

⁡

x

2

-

sin

2

⁡

x

2

sin

2

⁡

x

2

+

cos

2

⁡

x

2

=

1

-

tan

2

⁡

x

2

1

+

tan

2

⁡

x

2

=

1

-

t

2

1

+

t

2

.

Example.

The above formulae give from

∫

d

⁢

x

sin

⁡

x

the result

∫

d

⁢

x

sin

⁡

x

=

∫

1

+

t

2

2

⁢

t

⋅

2

⋅

1

1

+

t

2

⁢

𝑑

t

=

∫

d

⁢

t

t

=

ln

⁡

|

t

|

+

C

=

ln

⁡

|

tan

⁡

x

2

|

+

C

(which can also be expressed in the form

-

ln

⁡

|

csc

⁡

x

+

cot

⁡

x

|

+

C

; see the

goniometric formulas

).

Note 1.

The substitution (2) is sometimes called the ‘‘universal trigonometric substitution’’ (

http://planetmath.org/UniversalTrigonometricSubstitution

).  In practice, it often gives rational functions that are too complicated.  In many cases, it is more profitable to use other substitutions:

•

In the case

∫

R

⁢

(

sin

⁡

x

)

⁢

cos

⁡

x

⁢

d

⁢

x

the substitution

sin

⁡

x

=

t

is simpler.

•

Similarly, in the case

∫

R

⁢

(

cos

⁡

x

)

⁢

sin

⁡

x

⁢

d

⁢

x

the substitution

cos

⁡

x

=

t

is simpler.

•

If the integrand depends only on

tan

⁡

x

, the substitution

tan

⁡

x

=

t

is simpler.

•

If the integrand is of the form

R

⁢

(

sin

2

⁡

x

,

cos

2

⁡

x

)

,  one can use the substitution

tan

⁡

x

=

t

; then

cos

2

⁡

x

=

1

1

+

tan

2

⁡

x

=

1

1

+

t

2

,

sin

2

⁡

x

=

1

-

cos

2

⁡

x

=

t

2

1

+

t

2

,

d

⁢

x

=

d

⁢

t

1

+

t

2

.

Example.

The integration of

∫

d

⁢

x

cos

4

⁡

x

⁢

𝑑

x

is of the last case:

∫

d

⁢

x

cos

4

⁡

x

⁢

𝑑

x

=

∫

1

(

cos

2

⁡

x

)

2

⁢

𝑑

x

=

∫

(

1

+

t

2

)

2

⋅

d

⁢

t

1

+

t

2

=

∫

(

1

+

t

2

)

⁢

𝑑

t

=

t

3

3

+

t

+

C

=

1

3

⁢

tan

3

⁡

x

+

tan

⁡

x

+

C

.

Example.

The

integral

I

=

∫

d

⁢

x

cos

3

⁡

x

⁢

𝑑

x

=

∫

sec

3

⁡

x

⁢

d

⁢

x

is a peculiar case in which one does not have to use the substitutions mentioned above, as

integration by parts

is a simpler method for evaluating this integral. Thus,

u

=

sec

⁡

x

⇒

d

⁢

u

=

sec

⁡

x

⁢

tan

⁡

x

⁢

d

⁢

x

;

d

⁢

v

=

sec

2

⁡

x

⁢

d

⁢

x

⇒

v

=

tan

⁡

x

.

Therefore,

I

=

∫

sec

3

⁡

x

⁢

d

⁢

x

=

sec

⁡

x

⁢

tan

⁡

x

-

∫

sec

⁡

x

⁢

tan

2

⁡

x

⁢

d

⁢

x

=

sec

⁡

x

⁢

tan

⁡

x

-

∫

sec

⁡

x

⁢

(

sec

2

⁡

x

-

1

)

⁢

𝑑

x

=

sec

⁡

x

⁢

tan

⁡

x

-

I

+

∫

sec

⁡

x

⁢

d

⁢

x

,

and consequently

∫

d

⁢

x

cos

3

⁡

x

⁢

𝑑

x

=

1

2

⁢

(

sec

⁡

x

⁢

tan

⁡

x

+

ln

⁡

|

sec

⁡

x

+

tan

⁡

x

|

)

+

C

.

Note 2.

There is also the ‘‘universal hyperbolic substitution’’ for integrating rational functions of

hyperbolic sine

and cosine:

tanh

⁡

x

2

=

t

,

d

⁢

x

=

2

⁢

d

⁢

t

1

-

t

2

,

sinh

⁡

x

=

2

⁢

t

1

-

t

2

,

cosh

⁡

x

=

1

+

t

2

1

-

t

2

References

1

Л. Д. Кдрячев:

Математичецкии  анализ

.  Издательство
‘‘ВүсшаяШкола’’. Москва (1970).

Title

integration of rational function of sine and cosine

Canonical name

IntegrationOfRationalFunctionOfSineAndCosine

Date of creation

2013-03-22 17:05:15

Last modified on

2013-03-22 17:05:15

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

29

Author

pahio (2872)

Entry type

Topic

Classification

msc 26A36

Synonym

universal trigonometric substitution

Related topic

GoniometricFormulae

Related topic

SubstitutionForIntegration

Related topic

WeierstrassSubstitutionFormulas

Related topic

EulersSubstitutionsForIntegration

Related topic

ErrorsCanCancelEachOtherOut

Defines

universal hyperboloc substitution

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