Question

Integrate
$(4x+2)\sqrt{x^2+x+1}$

Answer 1

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

Let $t=x^2+x+1\Rightarrow dt=(2x+1)dx$

$=2\int (2x+1)\sqrt{x^2+x+1}dx$

$=2\int \sqrt{t}dt$

$=2\int t^{\frac{1}{2}}dt$

$=2\left[\frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right]+c$

$=2\times \frac{2}{3}{t}^{\frac{3}{2}}+c$ where $t=x^2+x+1$

$=\frac{4}{3}{(x^2+x+1)}^{\frac{3}{2}}+c$