Try integration by parts with
and
I need help integrating:
xarctan(x) / ((1+x^2)^2)
I did trig substitutions like this:
u=tan(theta)
du=sec^2(theta) d(theta)
So I ended up with:
Integral of (tan(theta)*arctan(tan(theta))*(sec^2(theta))) / ((1+tan^2(theta))^2)
Which simplifies into:
Integral of:
(Theta)(tan(theta))(sec^2(theta)) / (sec^2(theta))^2
The sec^2(theta)'s cancel so I have:
(theta)(tan(theta))(cos^2(theta)), which simplifies into:
(theta)(sin(theta))(cos(theta))
Is that right, and if so, how do I integrate it?
I took cos^2(theta) to (1/2)(1+cos(2theta)).
So I now have it down to:
-arctan(x)/(2(x^2+1)) + (1/2)*((1/2)theta + (1/4)sin (2theta))
So, with the subsitution x=tan(theta), theta=arctan(x), so this factors to:
(1/4)arctan(x) + (1/8)sin (2theta))
What do I do with the (1/8)sin(2theta)? It's not good to put arctan(x) in the sin, to have (1/8)sin(2arctan(x)), right?
I put it to 2sin(theta)cos(theta), so it's:
(1/4)arctan(x) + (1/4)sin(theta)cos(theta)
What's the best way to factor the last bit?