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How to integrate powers of tan and cot
functions
First powers
We have the standard formulae
Z
tan x dx = − ln | cos x| + C = ln | sec x| + C
and
(1)
Z
cot x dx = ln | sin x| + C .
(2)
Squares
Noting the derivatives
d
(tan x) = sec2 x
dx
and
d
(cot x) = − csc2 x
dx
and
cot2 x = csc2 x − 1 ,
(3)
and the identities
tan2 x = sec2 x − 1
we obtain
Z
Z
2
tan x dx =
(sec2 x − 1) dx = tan x − x + C
(4)
(5)
and
Z
cot2 x dx =
Z
(csc2 x − 1) dx = − cot x − x + C .
(6)
Higher powers
Reduction formulae, expressing integrals of powers of tan and cot functions in
terms of integrals of lower powers of the same functions, are easy to derive
using the identities (4) above – there’s no need Rto use integration by parts. The
are shown below, in full
for tann x dxRand in less detail for
Rderivations
R detail
n
n
cot x dx . Also remember that tan (ax + b) dx and cotn (ax + b) dx can
be evaluated by the method for integrating functions of linear expressions.
1
Z
Z
n
tann x dx
R
Reduction formula for
tann−2 x tan2 x dx
tan x dx =
Z
tann−2 x (sec2 x − 1) dx
Z
Z
n−2
2
=
tan
x sec x dx −
tann−2 x dx .
=
(7)
In the first integral in (7), use the substitution
u = tan x ,
then
Z
Z
tann−2 x sec2 x dx =
Thus (7) yields
Z
tann x dx =
du = sec2 x dx ;
so that
un−2 dx =
1
un−1
+C =
tann−1 x + C .
n−1
n−1
1
tann−1 x −
n−1
Z
tann−2 x dx .
(8)
Depending on whether n is odd or even,R repeated useR of the reduction
formula (8) will reduce the original integral to tan x dx or tan2 x dx , which
can be evaluated by the formulae (1) or (5) above, respectively. The process is
demonstrated in worked example no. 1.
Reduction formula for
Z
cotn x dx =
R
Z
cotn x dx
cotn−2 x cot2 x dx
Z
=
.
..
Z
cotn x dx = −
cot
n−2
1
n−1
2
Z
x csc x dx −
cotn−2 x dx
Z
cotn−1 x −
cotn−2 x dx ,
(9)
(10)
where the substitution u = cot x , du = − csc2 x dx
has been used to
evaluate the first integral in (9).
Depending on whether n is odd or even, Rrepeated use Rof the reduction
formula (10) will reduce the original integral to cot x dx or cot2 x dx , which
can be evaluated by the formulae (2) or (6) above, respectively. The process is
demonstrated in worked example no. 2.
Worked examples of integrating powers of tan and cot functions
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involving half-integer powers of quadratics”
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