Your steps are alright, just that the sketch of the graph is of tan x instead of tan-¹ x

If the graph is meant to be tan x indeed, then for the limits it should be tan x instead of tan-¹x

It was intended to be tan x, to show him that tan (pi/2) and tan (-pi/2) is infinite, so by converse, inverse tangent of this “infinite” is pi/2 and so on.

However, the integral has to be continuous (non-broken) so the limits of the integral has to fall within the principal values of -pi/2 and pi/2.

When x = π/2, tan-¹ x = tan-¹ (π/2) ≈ 1.004 rad
When x = -π/2, tan-¹ x = tan-¹ (-π/2) ≈ -1.004 rad


The correct expressions would be :

lim tan x = -∞
x→π/2 ⁺

lim tan x = ∞
x→π/2 ⁻

lim tan x = -∞
x→-π/2 ⁺

lim tan x = ∞
x→-π/2 ⁻

Whoops

I just realised I wrote tan inverse (pi/2)- = infinity when I meant tan (pi/2)- = infinity