Question

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

Answer 1

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

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

$put{x}^2+x+1=t$

$\left(2x+1\right)dx=dt$

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

$=\frac{2t^{\frac{3}{2}}}{\frac{3}{2}}+c$ $\because \int t^{n}dt=\frac{t^{n+1}}{n+1}+c$

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

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