Expert: Paul Klarreich - 12/8/2008

Question
Hi

I am stuck on an integration problem.  Your help would be most welcome.

Int[(arctanx)/x^2)]

Thanks for your time!

-Anson

Answer
Questioner:   Anson
Category:  Calculus
Private:  No
 
Subject:  Calculus
Question:  Hi

I am stuck on an integration problem.  Your help would be most welcome.

Int[(arctanx)/x^2)]

Thanks for your time!
............................
Hi, Anson,

Looks like a straightforward integration by parts.


{ arctan(x)
| ----------- dx
}    x^2

Let
                      dx
u = arctan(x),  du = --------
                    1 + x^2

dv = x^(-2) dx;  v = -1/x

Now the uv part:

     - arctan x
uv = ------------
         x

And the

 {
- | vdu =
 }     


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


Now use partial fractions:

    1          A     Bx + C
----------- = --- + --------
x(1 + x^2)     x    1 + x^2


1 = A + Ax^2 + Bx^2 + Cx

A = 1;  C = 0

A + B = 0;  B = -1

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

Now you can handle the rest, I think:
1/x integrates to a ln(x)

The second term integrates using  u = 1 + x^2.


P.S. Check out this web site:

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