Question

Evaluate: $\int (4x+2)\sqrt[3]{3x^2+x+1dx}$

Answer 1

$I=\int (4x+2)\sqrt[3]{3x^2+x+1}dx$

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

$\Rightarrow I=2\int (2x+1)\sqrt[3]{3x^2+x+1}=\int t^{1/3}dt$

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