Expert: Paul Klarreich - 4/18/2007

Question
an antiderivative for 1/(x^2-2x+2) is?
can you show me this problem because i can't get an answer

Answer
Questioner:   h
Category:  Calculus
 
Subject:  antiderivative
Question:  an antiderivative for 1/(x^2-2x+2) is?
can you show me this problem because i can't get an answer
.........................................
Hi, h,

It is a good idea to follow the instructions as closely as possible; here I am going to have to guess just what stage of calculus you have reached.  I shall assume that, since this is mid-April, you are doing Methods of Integration; chapter 9 in most Calculus texts.

Try this:

Complete the square in  x^2 - 2x + 2:

x^2 - 2x + 2 =
x^2 - 2x + 1 + 1
(x - 1)^1 + 1

So your integral is:

{      1
| ------------- dx
} (x - 1)^2 + 1

Now let  u = (x-1);  du = dx

{    du
| -------
} u^2 + 1

Now this looks like a trig-substitution.  Think of your right triangle with 1,u, and  sqrt(u^2 + 1) as the sides.

Let u = tan t;  du = sec^2 t dt
u^2 + 1 = tan^2 t + 1 = sec^2 t

Your integral is now:

{ sec^2 t dt   
| ---------- =
}   sec^2 t

{
| dt = t + C
}

Now  t = arctan(u) = arctan(x - 1) + C.