# Thread: Completing the Square in Integration

1. ## Completing the Square in Integration

When is it necessary to complete the square? What would be an example?

2. ## Re: Completing the Square in Integration

becomes very useful in trig substitution problems

$\displaystyle \int \frac{1}{\sqrt{x^2+bx+c}}$

$\displaystyle \int \frac{1}{\sqrt{(x+\frac{b}{2})^2+(c-\frac{b^2}{2^2})}}$

use

$\sqrt{(c-\frac{b^2}{2^2})}}\tan \theta = x+\frac{b}{2}$

3. ## Re: Completing the Square in Integration

One thing I'm seeing on You Tube is $\int \dfrac{dx}{u^{2} + a} = \dfrac{1}{a}\arctan\dfrac{u}{a} + C$

and you get into this form by completing the square. However, can you be lead in certain situations to other inverse trig functions?

4. ## Re: Completing the Square in Integration

There are a number of integration formulas for
$\int \frac{dx}{x^2+ a}$
$\int \frac{dx}{x^2- a}$
$\int \frac{dx}{\sqrt{x^2+ a}}$
$\int \frac{dx}{a- x^2}$
$\int \frac{dx}{x^2- a}$, etc.

All of those can be reached from more complicated integrals by "completing the square".