Integration by Parts: Int: x*arctan(x) dx

[MathHawk]

Because of circumstance (my desire to graduate in 5 years or less), I've been forced to attempt Calc 2 in 2 months time online over the summer. About 75% of it is going smoothly (compared with 105% or so of Calc 1).

Homework Statement

I'm to solve the indefinite integral: \int x * arctan(x) dx

Homework Equations

Integration by parts is done using: \intu dv = uv - \intv du

The Attempt at a Solution

It seems pretty obvious that u = arctan(x) and that dv = x. From this, du = \frac{1}{1+x^2} dx and v = \frac{1}{2}x².

Using the integration by parts formula:

\int x * arctan(x) dx = \frac{1}{2}x²arctan(x) - \frac{1}{2}\int\frac{x^2}{1 + x^2}dx



Now integration by parts must be used again. It seems obvious to select u = x², dv = \frac{1}{1 + x^2}. du = 2xdx, dv = arctan(x).



\int x * arctan(x) dx = \frac{1}{2}x²arctan(x) - \frac{1}{2} ( x²arctan(x) - 2 \intxarctan(x) ).

Simplified:

\int x * arctan(x) dx = 0 + \int x * arctan(x) dx.



While this is very true, it doesn't help me find the integral. Switching my u and dv in either use of the integration by parts formula hasn't yielded a solution for me in my attempts yet. Thank you in advance for your help .

 

[rock.freak667]

\frac{x^2}{x^2+1}=\frac{x^2+1 -1}{x^2+1}=1-\frac{1}{x^2+1}

 

[MathHawk]

You are my Bokonon, only you tell truths.

 

[MathHawk]

Thank you, in other words.