Integration of sec x and sec3 x

R
sec x dx – by trickery
sec x+tan x
The standard trick used to integrate sec x is to multiply the integrand by 1 = sec x+tan x
and then
2
substitute y = sec x + tan x, dy = (sec x tan x + sec x) dx.
Z Z Z Z
sec x+tan x sec2 x+sec x tan x dy
sec x dx = sec x sec x+tan x dx = sec x+tan x
dx = y
= ln |y| + C

= ln | sec x + tan x| + C

R
sec x dx – by partial fractions
R
Another method for integrating sec x dx, that is more tedious, but less dependent on a memorized
R
trick, is to convert sec x dx into the integral of a rational function using the substitution y = sin x,
dy = cos x dx and then use partial fractions.
Z Z Z Z Z
1 cos x cos x 1
sec x dx = cos x
dx = cos2 x
dx = 1−sin2 x
dx = 1−y 2
dy
Z h i Z h i h i
1 1 1 1 1 1 1
= 2 1+y + 1−y dy = 2 y+1 − y−1 dy = 2 ln |y + 1| − ln |y − 1| +C
y+1 
= 12 ln  y−1 + C = 12 ln  sin
sin x+1 
   
x−1 + C

To see that this answer is really the same as the one above, note that
(sin x+1)2 2
sin x+1
sin x−1 = = (sin
sin2 x−1
x+1)
− cos2 x
 sin x+1  1  (sin x+1)2  x+1)2 
⇒ 21 ln  sin 1
ln  (sin
   sin x+1 
x−1 = 2 ln − cos2 x
  = 2 = ln   = ln | tan x + sec x|
2 cos x cos x

sec3 x dx – by trickery
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The standard trick used to evaluate sec3 x dx is integration by parts with u = sec x, dv =
R

sec2 x dx, du = sec x tan x dx, v = tan x.

Z Z Z
3 2
sec x dx = sec x sec x dx = sec x tan x − tan x sec x tan x dx

Since tan2 x + 1 = sec2 x, we have tan2 x = sec2 x − 1 and

Z Z Z
sec x dx = sec x tan x − [sec x − sec x] dx = sec x tan x + ln | sec x + tan x| + C − sec3 x dx
3 3

where we used sec x dx = ln | sec x + tan x| + C. Now moving the sec3 x dx from the right hand
R R

side to the left hand side

Z
2 sec3 x dx = sec x tan x + ln | sec x + tan x| + C
Z
⇒ sec3 x dx = 12 sec x tan x + 12 ln | sec x + tan x| + C

for a new arbitrary constant C.

June 19, 2019 Integration of sec x and sec3 x Joel Feldman 1

 sec3 x dx – by partial fractions
R

Another method for integrating sec3 x dx, that is more tedious, but less dependent on trickery,
R

is to convert sec3 x dx into the integral of a rational function using the substitution y = sin x,
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dy = cos x dx and then use partial fractions.

Z Z Z Z Z Z
sec3 x dx = 1
cos3 x dx = cos x
cos4 x dx = cos x
[1−sin2 x]2
dx = 1
[1−y 2 ]2 dy = 1
[y 2 −1]2 dy

i2
Z h Z h i
1 1 1 1 1 2 1
= 2 y−1
− y+1
dy = 4 (y−1) 2 − (y−1)(y+1)
+ (y+1) 2 dy
Z h i
= 14 1
(y−1) 2 − 1
y−1 + 1
y+1 + 1
(y+1) 2 dy
h i
+ C = − 41 y22y−1 + 14 ln  y−1
 y+1 
= 14 − y−1 1
− ln |y − 1| + ln |y + 1| − y+1 1 +C
y  y+1 
= 21 1−y 1 1 sin x 1  sin x+1 
   
2 + 4 ln y−1 + C = 2 cos2 x + 4 ln sin x−1 + C

June 19, 2019 Integration of sec x and sec3 x Joel Feldman 2