When is it necessary to complete the square? What would be an example?
becomes very useful in trig substitution problems
$\displaystyle \displaystyle \int \frac{1}{\sqrt{x^2+bx+c}}$
$\displaystyle \displaystyle \int \frac{1}{\sqrt{(x+\frac{b}{2})^2+(c-\frac{b^2}{2^2})}}$
use
$\displaystyle \sqrt{(c-\frac{b^2}{2^2})}}\tan \theta = x+\frac{b}{2}$
One thing I'm seeing on You Tube is $\displaystyle \int \dfrac{dx}{u^{2} + a} = \dfrac{1}{a}\arctan\dfrac{u}{a} + C$
and you get into this form by completing the square. However, can you be lead in certain situations to other inverse trig functions?
There are a number of integration formulas for
$\displaystyle \int \frac{dx}{x^2+ a}$
$\displaystyle \int \frac{dx}{x^2- a}$
$\displaystyle \int \frac{dx}{\sqrt{x^2+ a}}$
$\displaystyle \int \frac{dx}{a- x^2}$
$\displaystyle \int \frac{dx}{x^2- a}$, etc.
All of those can be reached from more complicated integrals by "completing the square".