Question

What is the ∫1/x2+x+1 dx ?

Answer 1

You completed the square perfectly fine.
Now the integral looks like this: \(\large \int \frac{1}{(x+\frac{1}{2})^2+(\frac{\sqrt 3}{2})^2} dx\)

Answer 2

Why do you put 3/4 as the (3√2)^2 ?
\[ \int\limits_{}^{} \frac{\frac{ 4 }{ 3 }\ 1 }{ (\frac{ 4 }{ 3 }((x+\frac{ 1 }{ 2 })^{2} +\frac34)} \]
\[\frac{ 4 }{ 3 }\ \int\limits_{}^{} \frac{ 1 }{ (\frac{ 4 }{ 3 }(x+\frac{ 1 }{ 2 })^{2} +1)} \]

And what is next ?

Answer 3

This 3/4 has been written as (sqrt(3/4))^2 because we know a formula which says...

Ofcourse with a constant :-)