Question

Integral de x^2 vezes a raiz de (3x+2) definida em 2 e 0

Answer 1

1) Cambio de variable
$u^2=3x+2\to 2udu=3dx\to \boxed{dx =\dfrac{2u}{3}du}\\ \\ \\ x=\dfrac{u^2-2}{3}\to x^2=\dfrac{(u^2-2)^2}{9}\to \boxed{x^2=\dfrac{u^4-4u^2+4}{9}}$

2) cambio de límites
            $\text{sim }x=0\text{ ent\~ao }u=\sqrt{2}\\ \\ \text{sim }x=2\text{ ent\~ao }u=2\sqrt{2}$

3) Cálculo de la integral resultante
                            
                         $\displaystyle \int_{0}^2x^2\sqrt{3x+2}dx=\int_{\sqrt{2}}^{2\sqrt{2}}\dfrac{u^4-4u^2+4}{9}\cdot \dfrac{2u^2}{3}du\\ \\ \\ \int_{0}^2x^2\sqrt{3x+2}dx=\dfrac{2}{27}\int_{\sqrt{2}}^{2\sqrt{2}}u^6-4u^4+4u^2\;du\\ \\ \\ \int_{0}^2x^2\sqrt{3x+2}dx=\dfrac{2}{27}\left.\left(\dfrac{u^7}{7}-\dfrac{4u^5}{5}+\dfrac{4u^3}{3}\right)\right|_{\sqrt{2}}^{2\sqrt{2}}\\ \\ \\ \int_{0}^2x^2\sqrt{3x+2}dx=\dfrac{2}{27}\left(\dfrac{6784\sqrt{2}}{105}\right)\\ \\ \\ \boxed{\int_{0}^2x^2\sqrt{3x+2}dx=\dfrac{13568\sqrt{2}}{2835}}$