Integral of tan3 (x)
Use what you have learned to integrate the function tan3 (x).
Solution
This is a relatively simple integration; the method �described below uses a sub­
stitution and the properties sec2 x = 1 + tan2 x and tan x dx = − ln | cos x| + c.
�
tan3 x dx =
�
tan x tan2 x dx
(use an identity to reduce degree)
�
tan x(sec2 x − 1) dx
�
�
2
=
tan
x
sec
x
dx
−
tan x dx
� �� � � �� �
=
u
du
�
=
=
=
u du − (− ln | cos x| + c)
1 2
u + ln | cos x| + c
2
1
tan2 x + ln | cos x| + c
2
1
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18.01SC Single Variable Calculus��
Fall 2010 ��
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