whats the Integral of S x *arctan x dx?

Question

I had chosen:

u = arctan x dv = xdx
du = 1/(x^2 + 1) dx v = x^2/2

= xarctanx - 1/2 * S x^2 / (x^2 + 1) dx //applying udv - Svdu
= xarctanx - 1/2 * S 1 - 1/(x^2 + 1) dx //applied long division
= xarctanx - 1/2 [x - arctanx] //took integral
= xarctanx - 1/2x - 1/2arctanx

Is this right?

Answer 1

Close. If you let

u = arctan(x), dv = x dx
du = 1/(x^2 + 1) dx, v = (1/2)x^2

Your integration by parts should be

uv - Integral (v du)

And your first line should have been

(1/2)x^2 arctan(x) - Integral ( (1/2)x^2 * 1/(x^2 + 1) ) dx

And I'm more than convinced you can do the problem at this point. Just that one little error.