Question

A integral indefinida de ∫ x^2√ᵪ³-1 dx é

Answer 1

Boa tarde Andrea

∫ x² * √(x³ - 1) * dx 

fazendo 

u = x³ - 1 , du = 3x²*dx

1/3 ∫ √u * du = 1/3 * 2*u^(3/2)/3 

∫ x² * √(x³ - 1) * dx = 2/9 * (x³ - 1)^(3/2) + C 

Answer 2

$\mathrm{\int x^2\sqrt{x^3-1}\ dx}\\\\ \mathrm{u=x^3-1\ \ \|\ \ \dfrac{du}{dx}=3x^2\ \to\ \dfrac{du}{3}=x^2.dx}\\\\ \mathrm{\int x^2\sqrt{x^3-1}\ dx=\int \dfrac{1}{3}\sqrt{u}\ du=\dfrac{1}{3}\int u^{\frac{1}{2}}\ du=}\\\\ \mathrm{=\dfrac{1}{3}\dfrac{u^{\frac{1}{2}+1}}{\frac{1}{2}+1}=\dfrac{1}{3}\dfrac{u^{\frac{3}{2}}}{\frac{3}{2}}=\dfrac{2u^{\frac{3}{2}}}{9}=\boxed{\mathbf{\dfrac{2(x^3-1)^{\frac{3}{2}}}{9}+C}}}$