Chapter 4.3, Problem 1E

Interpretation Introduction

Interpretation:

To evaluate the integral ${\text{T}}_{\text{bottleneck}}\text{ = }{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{\text{dx}}{{\text{r}}^{\text{2}}{\text{+x}}^{\text{2}}}}$ by using identity ${\text{1+ tan}}^{\text{2}}\text{θ}={\text{sec}}^{\text{2}}$ and show that ${\text{T}}_{\text{bottlenock}}\text{= }\frac{\text{π}}{\sqrt{\text{r}}}$.

Concept Introduction:

A collision and disappearance of two equilibria in dynamical systems is a saddle-node bifurcation.

To solve the given problem, we need the trigonometric identity as ${\text{1+ tan}}^{\text{2}}\text{θ}={\text{sec}}^{\text{2}}$.