Question

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

Answer 1

Let $x^2+x+1=t$
$\Rightarrow (2x+1)dx=dt$
$\int (4x+2)\sqrt{x^2+x+1}dx$
$=\int 2\sqrt{t}dt$
$=2\int \sqrt{t}dt$
$=2\left(\frac{t^{\frac{3}{2}}}{\frac{3}{2}}\right)+C$
$=\frac{4}{3}(x^2+x+1{)}^{\frac{1}{2}}+C$