Why Desmos is giving two different answers for integral of the form $$I=\int_{0}^{\pi}\left(\tan^{-1}(\cot(mx)\right)^2\:dx,m \in \mathbb{N}$$
Firstly, I used the substitution $mx=\theta$. We get $dx=\frac{d\theta}{m}$, So $$I=\frac{1}{m}\int_{0}^{m\pi}\left(\tan^{-1}(\cot \theta)\right)^2\:d\theta$$
Now since the Fundamental Period of $\cot \theta$ is $\pi$, the integral converts to
$$I=\frac{1}{m}\times m \times \int_0^{\pi}\left(\tan^{-1}(\cot \theta)\right)^2\:d\theta$$
Now when $0<\theta<\pi$, we have $$\tan^{-1}(\cot \theta)=\frac{\pi}{2}-\theta$$ Thus, we get $$I=\int_0^{\pi}\left(\frac{\pi}{2}-\theta\right)^2\:d\theta=\frac{\pi^3}{12}=2.5838...$$
Now what I did is, I picked natural numbers $m=2022,2023$ and posted these on Desmos. Here is the Snapshot showing different results.
Any Comments?