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Prof: Stanley
Dept: mathematics statistics and actuarial
science
How can the integrate powers of sec
and csc functions
performed?
First powers
We have the standard formulae
¸
csc x dx = ln | csc x − cot x| + C
and

(1)

¸
sec x dx = ln | sec x + tan x| + C
...


(5)

Higher powers
We use integration by parts to obtain reduction formulae, which express
integrals of powers of csc and sec functions in terms of integrals of lower powers
¸
of the same functions
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Also remember that
cscn(ax + b) dx and
¸
secn(ax + b) dx can be evaluated by the method for integrating functions of
linear expressions
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(n − 2) cscn−3x (− csc x cot x) dx

2

(integrating by parts and using (4))
¸
= − cot x cscn−2x − (n − 2)
cscn−2x cot2x dx
¸
n−2
= − cot x csc x − (n − 2)
cscn−2x (csc2x − 1) dx
cot2x = csc2x − 1 )
¸
¸
= − cot x cscn−2x − (n − 2)
cscnx dx + (n − 2)
cscn−2x dx
...
Bringing these terms together on the left-hand side, (6) becomes
¸

¸
cscnx dx = − cot x cscn−2x + (n − 2)
cscn−2x dx
¸
¸
1 cot x cscn−2x + n − 2

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The process is
demonstrated in example no
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Reduction formula for
¸

¸

secnx dx
¸

secnx dx =
=

sec2x secn−2x dx
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n−2
tan x sec x −
(tan x)
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¸
¸

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(n − 1)
¸
¸
1
n−2

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secnx dx =
tan x secn−2x +
secn−2x dx
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The process is demonstrated in
example no
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