# Thread: Completing the perfect square and tables of integration

1. ## Completing the perfect square and tables of integration

I had to Complete the perfect square and use the tables of integration to integrate the following:
∫ 1 / (√x^2 – 3x + 6) dx

What I have done:
Completing the perfect square we have;
x^2 – 3x + 6 = x^2 – 3x + 2.25 + 6 - 2.25 = (x – 1.5)^2 +3.75
if we set u = x – 1.5, then du = dx, the integral in terms of u is
∫ 1 / √(u^2 – 2.25) du

We then have
∫ 1 / √(u^2 – 2.25) du = ln|u +√u^2 – 2.25|,

And the answer in terms of x is
ln|x + 2 +√(u^2 – 2.25)| + C

Have I done this correctly?

2. Originally Posted by mellowdano
I had to Complete the perfect square and use the tables of integration to integrate the following:
∫ 1 / (√x^2 – 3x + 6) dx

What I have done:
Completing the perfect square we have;
x^2 – 3x + 6 = x^2 – 3x + 2.25 + 6 - 2.25 = (x – 1.5)^2 +3.75
if we set u = x – 1.5, then du = dx, the integral in terms of u is
∫ 1 / √(u^2 – 2.25) du

We then have
∫ 1 / √(u^2 – 2.25) du = ln|u +√u^2 – 2.25|,

And the answer in terms of x is
ln|x + 2 +√(u^2 – 2.25)| + C

Have I done this correctly?
Shouldn't the integral become after substitution

$
\int \frac{1}{\sqrt{u^2 + 3.75}}\,du
$
?

3. the way i'd complete the square is the following:

$\int{\frac{dx}{\sqrt{{{x}^{2}}-3x+6}}}=2\int{\frac{dx}{\sqrt{4{{x}^{2}}-12x+24}}}=2\int{\frac{dx}{\sqrt{{{(2x-3)}^{2}}+15}}},$ now put $2x-3=\sqrt{15}t.$

4. Originally Posted by Danny
Shouldn't the integral become after substitution

$
\int \frac{1}{\sqrt{u^2 + 3.75}}\,du
$
?

so by making those changes i would have

Completing the perfect square we have;
x^2 – 3x + 6 = x^2 – 3x + 2.25 + 6 - 2.25 = (x – 1.5)^2 +3.75
if we set u = x – 1.5, then du = dx, the integral in terms of u is
∫ 1 / √(u^2 + 3.75) du

We then have
∫ 1 / √(u^2 + 3.75) du = ln|u +√u^2 + 3.75|,

And the answer in terms of x is
ln|x + 2 +√(u^2 + 3.75)| + C

5. I haven't learnt to complete square the way you have, I am trying to follow a similar example in my book. I have posted a correction that Danny pointed out.