MAT142 Supplement- Technique of integrations-
February 17, 2014
Two different strategies to solving tan 4 xdx .
Method 1: Using the trigonometric identity tan 2 x 1 sec 2 x , solve for tan 2 x which is
sec 2 x -1. We can separate the integrand into
tan 4 xdx
tan 2 x tan 2 xdx
(sec2 x 1)(sec2 x 1)dx
Expand and separate into 3 integrals to obtain sec4 xdx 2 sec2 xdx
to use integration by parts for sec4 xdx
let
u
du
sec 2 x
2sec 2 x tan xdx
dx , we need
dv sec 2 xdx
v
tan x
sec4 xdx sec2 x tan x 2 tan 2 x sec2 xdx use the trig identity tan 2 x 1 sec 2 x again
to replace tan 2 x with sec 2 x -1 to get
sec 4 xdx sec2 x tan x 2 tan 2 x sec 2 xdx
sec 2 x tan x 2 (sec 2 x 1) sec 2 xdx
sec 2 x tan x 2 sec 4 xdx 2 sec 2 xdx
Notice that there is a sec4 xdx on the right side with a -2 as its coefficient, so we can
move that to the left side getting
3 sec4 xdx = sec2 x tan x 2 sec2 xdx divide both sides by 3 to get
1
2
sec2 xdx
sec4 xdx = sec2 x tan x
3
3
Combining the above and sec4 xdx 2 sec2 xdx
1
2
sec2 xdx 2 sec2 xdx
tan 4 xdx = sec2 x tan x
3
3
rest, giving us
dx we get
dx , now we can integrate the
1
2
sec2 xdx 2 sec2 xdx dx
tan 4 xdx = sec2 x tan x
3
3
1
4
tan x x C
tan 4 xdx = sec2 x tan x
3
3
d 1 2
4
( sec x tan x
tan x x C ) tan 4 x . I will leave this for you to do.
Verify that
dx 3
3
Method 2: separate the integrand into two tan 2 x and replace one of them by sec 2 x -1, giving
us the following:
MAT142 Supplement- Technique of integrations-
tan 4 xdx
tan 2 x tan 2 xdx
tan 2 x sec 2 xdx
tan 3 x
3
February 17, 2014
tan 2 x(sec2 x 1)dx
tan 2 xdx use substitution for the first integral
(sec2 x 1)dx
tan 3 x
tan x x C
3
d tan 3 x
Verify again that
(
tan x x C )= tan 4 x
dx
3
Note: There may be other ways to integrate this problem but this is only an example of the
importance of recognizing which strategy works better or faster gets us to the solution. Notice
when I did this problem in class, I was stuck on sec4 xdx . Then I tried method 2 and found that
was an easier way. I am not saying that you have to always find the easier way but that it is a
consideration. And if you went the long way, as long as you can complete the solution, it is fine.
Our goal is of course to find the most efficient technique to solve a problem.