Chapter 13.3, Problem 63E

Improper integrals Improper integrals arise in polar coordinates when the radial coordinate r becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way:

     ${\displaystyle {\int }_{\alpha }^{\beta }{\displaystyle {\int }_a^{\infty }f\left(r,\theta \right)}}rdrd\theta =\underset{b\to \infty }{\lim }{\displaystyle {\int }_{\alpha }^{\beta }{\displaystyle {\int }_a^{b}f\left(r,\theta \right)}}rdrd\theta .$

Use this technique to evaluate the following integrals.

63.     ${\displaystyle {\int }_0^{\pi /2}{\displaystyle {\int }_1^{\infty }\frac{\cos \theta }{r^3}}rdrd\theta }$