Question

Evaluate $\int \frac{2x-3}{\sqrt{x^2+x+1}}dx$

Answer 1

Given that:

$\Rightarrow \int \frac{2x-3}{\sqrt{x^2+x+1}}dx$

$\Rightarrow \int (\frac{2x+1}{\sqrt{x^2+x+1}}-\frac{4}{\sqrt{x^2+x+1}})dx$

$\Rightarrow \int \frac{2x+1}{\sqrt{x^2+x+1}}-\int \frac{4}{\sqrt{x^2+x+1}}dx$

Let $x^2+x+1=t$ in First part, than $(2x+3)dx=dt$

$\Rightarrow \int \frac{1}{\sqrt{t}}dt-\int \frac{4}{\sqrt{x^2+x+\frac{1}{4}+\frac{3}{4}}}dx$

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

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