<s>[INST] Find $\int\frac{x+1}{x^2+x+1}dx$ Find
$\int\frac{x+1}{x^2+x+1}dx$
$\int \frac{x+1dx}{x^2+x+1}=\int \frac{x+1}{(x+\frac{1}{2})^2+\frac{3}{4}}dx$
From here I don't know what to do.Write $(x+1)$ = $t$?
This does not work.Use partial integration?I don't think it will work here.
And I should complete square then find.
 [/INST] Write $x+\frac{1}{2} = t.$ Then,
$$\frac{x+1}{\left(x+\frac{1}{2}\right)^2+\frac{3}{4}} = \frac{x+\frac12 + \frac12}{\left(x+\frac{1}{2}\right)^2+\frac{3}{4}} = \frac{t}{t^2+\frac{3}{4}} + \frac{\frac12}{t^2+\frac{3}{4}}.$$
Can you integrate these two terms now?
</s>