Expert: Paul Klarreich - 7/27/2009

Question


Can you tell me, can i use integration by part or u sub for 2secX/2+tanX, or both, i cant find secx in the integral tables anywhere. thanks for your recent answers.

Dave


Answer
Questioner: Dave
Country: United Kingdom
Category: Calculus
Private: No
Subject: integration
Question:

Can you tell me, can i use integration by part or u sub for 2secX/2+tanX, or both, i cant find secx in the integral tables anywhere. thanks for your recent answers.

Dave
............................
Hi, Dave,

I am not sure exactly what you are writing:  Do you mean:

{
| ( 2 sec(x/2) + tan x) dx  ??
}

If so, then there really is no problem.  Just handle each term by itself.  The first term is just 4 sec t, after an obvious substitution of  t = x/2.

Then you can look it up.  Try an integral table from the United States -- we have it.

Just joking, of course, but it is a pretty standard integral.  If you really can't find it, use this trick:

{
| sec(t) dt  =
}

{ sec t (sec t + tan t) dt
| ------------------------- =
}  (sec t + tan t)

{ sec^2(t) + sec t tan t
| ----------------------- dt =
}  (sec t + tan t)

Now, amazingly, the top is the derivative of the bottom and another u-substitution does it.

As to the second term.

{
| tan x dx =
}

{  sin x
| ------- dx =
}  cos x

And let  u = cos x

BTW, the 'lazy' comment was intended to point out that if you leave out the 'dx' in the integral you might miss the working-out of the u-substitution.  'Lazy' is not a derogatory term when applied to mathematicians.

One more suggestion:  Try this site:

integrals.wolfram.com/index.jsp

for a quick-and-dirty answer.  It won't tell you how it got it, but the answer might be a good hint.