Integral of sec4 x
We now know the following facts about integration of trig functions:
sec x =
1
cos x
tan x =
sin x
cos x
sin2 x + cos2 x = 1
csc x =
1
sin x
ctnx =
cos x
sin x
sec2 x = 1 + tan2 x
d
dx
d
dx
tan x = sec2 x
d
dx
d
dx
sin x = cos x
R
sec x = sec x tan x
cos x = − sin x
R
tan x dx = − ln(cos(x)) + c
sec x dx = ln(sec x + tan x) + c
We’ve also seen several useful integration techniques, including methods for
integrating any function of the form sinn x cosm x. At this point we have the
tools needed to integrate most trigonometric polynomials.
Z
Example:
sec4 x dx
We can get rid of some factors of sec x using the identity sec2 x = 1 + tan2 x.
This is a particularly good idea because sec2 x is the derivative of tan x.
Z
sec4 x dx =
Z
Z
=
(1 + tan2 x) sec2 x dx
(1 + tan2 x) sec2 x dx
Using the substitution u = tan x, du = sec2 x dx, we get:
Z
Z
sec4 x dx =
(1 + u2 )du
=
Z
sec4 x dx =
u3
+c
3
tan3 x
tan x +
+c
3
u+
1