Question

Integrate the following functions.
$\int (4x+2)\sqrt{x^2+x+1}dx.$

Answer 1

Let $I=\int (4x+2)\sqrt{x^2+x+1}dx$
Let $x^2+x+1=t$
On differentiating w.r.t.x, we get $2x+1=\frac{dt}{dx}\Rightarrow dx=\frac{dt}{(2x+1)}$
$\therefore I=\int (4x+2)\sqrt{t}\frac{dt}{(2x+1)}=\int 2(2x+1)\sqrt{t}\frac{dt}{(2x+1)}=2\int \sqrt{t}dt=2\frac{t^{\left(\frac{1}{2}\right)+1}}{\left(\frac{1}{2}\right)+1}+C=\frac{4}{3}(x^2+x+1{)}^{\frac{3}{2}}+C$