1. The problem statement, all variables and given/known data [tex] \int x \arctan x \, dx [/tex] 3. The attempt at a solution By parts, [tex] u = \arctan x[/tex] [tex] dv = x dx[/tex] [tex] du = \frac{dx}{x^2+1}[/tex] [tex]v = \frac{x^2}{2} [/tex] [tex] \int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{1}{2} \int \frac{x^2}{x^2+1} \, dx [/tex] Again...by parts [tex] u = x^2 [/tex] [tex] dv = \frac{dx}{x^2+1} [/tex] [tex] du = 2x dx [/tex] [tex] v = arc tan x [/tex] [tex]\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{x^2}{2}\arctan x - \int x \arctan x \, dx [/tex] I back to the beginning, what did wrogn? [tex]\int x \arctan x \, dx = - \int x \arctan x \, dx [/tex]
[tex]\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{x^2}{2}\arctan x - \int x \arctan x \, dx [/tex] Add [tex]\int x \arctan x \, dx[/tex] to both sides, then solve for the integral, assuming your work is correct.
you mean like this? is the same, i back to the beginign [tex]\int x \arctan x \, dx +\int x \arctan x \, dx = \frac{x^2}{2}\arctan x - \frac{x^2}{2}\arctan x - \int x \arctan x \, dx +\int x \arctan x \, dx[/tex] [tex]2\int x \arctan x \, dx = 0[/tex]