Question

Find the value of $\int \frac{2x+1}{\sqrt{x^2+x+1}}dx$

Answer 1

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

$x^2+x+1=t$ (Let)

Differentiate with respect to $x$

$\frac{d}{dx}(x^2+x+1)=\frac{dt}{dx}$

$2x+1=\frac{dt}{dx}$

$(2x+1)dx=dt$

$\therefore I=\int \frac{dt}{\sqrt{t}}=\int t^{-\frac{1}{2}}dt$

$I=\frac{t^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}+c$

$I=\frac{t^{\frac{1}{2}}}{\frac{1}{2}}+c$

$I=2\sqrt{t}+c$

$I=2\sqrt{x^2+x+1}+c$