# integral transforms – Regularization of the area under hyperbola

So, I am trying to find the regularized value of the divergent integral $$I=int_1^infty sqrt{x^2-1}dx$$. Since the area of $$int_0^1 sqrt{1-x^2}dx=fracpi4$$, I wonder whether the area under hyperbola would be interesting as well.

So, I applied my method that uses Laplace transform $$mathcal{L}_t(t f(t))(x)$$ so to convert divergent integrals into equivalent ones.

This way I obtained a family of equivalent integrals:

$$int_1^infty sqrt{x^2-1}dx=int_0^infty frac{K_2(x)}{x}dx=int_0^infty left(x-frac{1}{2 x}right) dx =int_0^infty left(frac{2}{x^3}-frac{1}{2 x}right)dx$$ Now, we already know that $$int_0^infty frac1xdx$$ regularizes to $$gamma$$ while $$int_0^infty frac1{x^3}dx$$ and $$int_0^infty xdx$$ regularize to zero, so integral $$I$$ regularizes to $$-fracgamma2$$.

This is a very interesting result.

That said, I wonder whether I could somehow find the regularized value of the area under conjugare hyperbola? The Laplace transform gives a very complicated function….

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