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## Homework Statement

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

## Homework Equations

Integration by parts is done using: [tex]\int[/tex]u dv = uv - [tex]\int[/tex]v du

## The Attempt at a Solution

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

^{2}.

Using the integration by parts formula:

[tex]\int[/tex] x * arctan(x) dx = [tex]\frac{1}{2}[/tex]x

^{2}arctan(x) - [tex]\frac{1}{2}[/tex][tex]\int[/tex][tex]\frac{x^2}{1 + x^2}[/tex]dx

Now integration by parts must be used again. It seems obvious to select u = x

^{2}, dv = [tex]\frac{1}{1 + x^2}[/tex]. du = 2xdx, dv = arctan(x).

[tex]\int[/tex] x * arctan(x) dx = [tex]\frac{1}{2}[/tex]x

^{2}arctan(x) - [tex]\frac{1}{2}[/tex] ( x

^{2}arctan(x) - 2 [tex]\int[/tex]xarctan(x) ).

Simplified:

[tex]\int[/tex] x * arctan(x) dx = 0 + [tex]\int[/tex] 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 .