Math 113
The W Substitution
The W substitution enables an rational function of the regular six trigonometric functions to
b in using the metho of partial fractions. It uses the substitution of
u = tan
x
2
. (1)
The full metho are substitutions for the v of dx , sin x , cos x , tan x , csc x , sec x , and cot x . Using
the iden tan
2
θ + 1 = sec
2
θ, the deriv of (1) is
du =
1
2
sec
2
x
2
dx =
1
2
h
1 + tan
2
x
2
i
dx =
1
2
1 + u
2
dx.
It follo that
dx =
2 du
1 + u
2
. (2)
T deriv the substitutions for sin x and the
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
1
u
√
1 + u
2
x
2
Figure 1: Reference triangle for u = tan(
x
2
)
other trigonometric substitutions, refer to figure
1 and use the double angle iden for sin x
and cos x . The double angle identit for sin x is
sin x = 2 sin
x
2
cos
x
2
and for cos x , the double angle iden is
cos x = cos
2
x
2
− sin
2
x
2
.
The substitution for sin x is
sin x = 2 sin
x
2
cos
x
2
= 2
u
√
1 + u
2
1
√
1 + u
2
=
2u
1 + u
2
(3)
Similarly for cos x , it is
cos x = cos
2
x
2
− sin
2
x
2
=
1
√
1 + u
2
2
−
u
√
1 + u
2
2
=
1 − u
2
1 + u
2
(4)
By using (3) and (4), the substitutions for tan x , csc x , sec x , and cot x is
tan x =
sin x
cos x
=
2u
1+u
2
1−u
2
1+u
2
=
2u
1 − u
2
csc x =
1
sin x
=
1 + u
2
2u
cot x =
1
tan x
=
1 − u
2
2u
sec x =
1
cos x
=
1 + u
2
1 − u
2
(5)
Note that the resulting equations for all 6 trigonometric functions, along with dx all are simple
p in u . Hence, integrals of rational functions of trigonometric functions can be solved using
partial fractions. In summary
u = tan
x
2
sin x =
2u
1 + u
2
csc x =
1 + u
2
2u
tan x =
2u
1 − u
2
dx =
2 du
1 + u
2
cos x =
1 − u
2
1 + u
2
sec x =
1 + u
2
1 − u
2
cot x =
1 − u
2
2u
