Question

Solve $\int \frac{1}{x\sqrt{x^2+x+1}}dx$

Answer 1

Given

$\int \frac{1}{x\sqrt{x^2+x+1}}dx$

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

$Put\frac{1}{x}=t,than\frac{1}{x^2}dx=-dt$

Now,

$\int \frac{-1}{\sqrt{t^2+t+1}}dt$

$\int \frac{-1}{\sqrt{t^2+t+\frac{1}{4}+\frac{3}{4}}}$

$\int \frac{-1}{\sqrt{{(t+\frac{1}{2})}^2+\frac{3}{4}}}$

$\Rightarrow -\log t+\sqrt{t^2+\frac{3}{4}}+constant$

$\Rightarrow -\log (\frac{1}{x}+\sqrt{\frac{1}{x^2}+\frac{3}{4}})+constant$