According to the Wikipedia, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
The motivation behind the Cauchy principal value has never been justified for me. Why should we find principal value of the integrals? Does not this let us to define principal value of every arbitrary non-convergent integral? For instance we can define principal value for integrals like $$\int_{-\infty}^{\infty}x\,\mathrm{d}x$$ or any other arbitrary integral.
I would like to know why cauchy principal value exists and what is it trying to remedy?
