Expert: Paul Klarreich - 8/24/2009

Question

I am a senior in AP Calculus BC and I am doing my summer assignment. I got stuck on this problem

Evaluate the indefinite integral:

integrand 1/ (x^2)(e^(2/x)) dx

I'm not even entirely sure how to start the problem.
I've tried looking up how to do it on various sites but none of them really made sense to me.
I also tried looking through my AP Calc AB notes, but couldn't find anything like it.
I don't understand what to do with the e.

Answer
Questioner: Courtney
Country: United States
Category: Calculus
Private: No
Subject: Indefinite Integral
Question:
I am a senior in AP Calculus BC and I am doing my summer assignment. I got stuck on this problem

Evaluate the indefinite integral:

integrand 1/ (x^2)(e^(2/x)) dx

I'm not even entirely sure how to start the problem.
I've tried looking up how to do it on various sites but none of them really made sense to me.
............................................
That is because you don't know the good ones.  Try the INTEGRATOR at

http://integrals.wolfram.com/index.jsp

The INTEGRATOR is to integrals as the TERMINATOR is to... well, you know what -- it just smashes them.

Now your integral is not exactly clear.  If you meant to write:

{      dx
| ------------
} x^2 e^(2/x)

then the Integrator gives this answer:

   1
----------
2 e^(2/x)

which means it really must be easy.  If you do this:

Let  u = 2/x,  then  du = -2/x^2  dx,  and  dx = - x^2 du / 2

{      dx
| ------------ =
} x^2 e^(2/x)

{   - x^2 du
| ------------ =
} 2 x^2 e^u

{   - du
| ------------ =
}   2 e^u

{
| -(1/2) e^-u du =  + 1/2 e^-u , etc.
}

I also tried looking through my AP Calc AB notes, but couldn't find anything like it.

............................
I think it's a Calc BC problem.  (just one of my jokes.)