In the Pi Manifesto we had some integrals that converged to [tex]\pi[/tex]:

[tex]\int_{-\infty}^\infty \operatorname{sech}(x)\,dx = \pi.[/tex]

[tex]\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\,dx = \pi.[/tex]

[tex]\int_{-\infty}^\infty \frac{1-\cos x}{x^2}\,dx = \pi.[/tex]

[tex]\int_{-\infty}^\infty \frac{\sin x}{x}\,dx = \pi.[/tex]

[tex]\int_{-\infty}^\infty \frac{\sin^2 x}{x^2}\,dx = \pi.[/tex]

[tex]\int_{-\infty}^\infty \frac{1}{1+x^2}\,dx = \pi.[/tex]

It's intriguing to me that each of those are exactly equal to [tex]\pi[/tex]! Are there any nice examples that are exactly equal to [tex]\tau[/tex]? (That is, without multiplying the above ones by 2 or sticking 1/2's throughout somewhere).