Question

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

Answer 1

Integration by parts is a good place to start.

Let `y=uv`. Then `intudv = uv - intvdu` - 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=x` and `u=arctan(x)`. Hence:
`(dv)/dx = 1 => dv=dx`
`(du)/dx = 1/(1+x^2) => du = dx/(1+x^2)`

Now, we simply substitute these values in to our formula:
`intudv = uv - intvdu`
`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.