How do you find the integral of f(x) = arctan(x)f(x)=arctan(x) ?

1 Answer
Sep 6, 2014

Integration by parts is a good place to start.

Let y=uvy=uv. Then intudv = uv - intvduudv=uvvdu - this is how to integrate by parts.

We need to choose a function u that is easily differentiated, and a function v that is easily antidifferentiated. For this, let's choose v=xv=x and u=arctan(x)u=arctan(x). Hence:
(dv)/dx = 1 => dv=dxdvdx=1dv=dx
(du)/dx = 1/(1+x^2) => du = dx/(1+x^2)dudx=11+x2du=dx1+x2

Now, we simply substitute these values in to our formula:
intudv = uv - intvduudv=uvvdu
therefore
intarctan(x)dx = arctan(x)*x - intx*dx/(1+x^2)

From here, we can antidifferentiate the far right-hand term quite easily, and come up with an answer.