Question

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

Answer 1

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

Let $x^2+x+1=z$

So, $(4x+2)dx=2dz$

Thus,

$I=\int 2\sqrt{z}dz$

$I=\frac{4z^{3/2}}{3}+c$

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