x is the angle whose sin is y. We draw a picture, and see that:
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcwRfdqY96DPhzPPdms4cQLlKmXVW1MaC8kuPbLgL01aUxNWqMHGaRGCwp7EpCHFVvx5oQC43kCJxJDj9fGmSYHhrOpAEtzDxGBZo5odYgXlXEshtd_-sFPqSpVtij50Ars4NyV9yajfyZ/s320/Screen+shot+2011-01-15+at+2.43.30+PM.png)
and
since we're dealing with the inverse function
sin-1 y
:slope of inverse = 1 / slope of original function
Since
sin-1 y + cos-1 y = π/2
, if:And our goal, remembering from here:
So, if we integrate
1/(1 + y2) dy
, we get tan-1 y
. We're going to use that.The geometric series is:
How do we derive this? One way (not legal) is to assume that the series really does converge to a sum S
Or we can check it by just multiplying out:
Replace x by -x2
If we integrate both sides, the rhs is
tan-1 x
. And the lhs is:The magic: if we let
x = 1
, then:Another series for π, how cool is that?