Expert: Paul Klarreich - 3/6/2006

Question
What is the Integration of

Sec x dx

Answer
Hi, Ibrahim,

You wrote:

Question:  What is the Integration of Sec x dx
------------------------------
This is a well-known trick.  (For you it will be well-known after you read this.)

Multiply and divide by  sec x + tan x:

{
| sec x dx =
}

{ sec x(sec x + tan x)
| --------------------- dx
}  (tan x + sec x)

[Note the tricky reversal of the terms on the bottom.]

{ sec^2 x + sec x tan x
| --------------------- dx
}    tan x + sec x

Now the top is exactly the derivative of the bottom.  So the simple substitution  u = tan x + sec x  makes the integral into:

{  1
| --- du = ln | u | = ln | sec x + tan x | + C, if you like.
}  u