Question

Resolva a integral I = ∫ 3 x 2 √ 1 + x 3 d x e marque a opção correta:

Answer 1

Resposta:

$\frac{2}{3} (1+x^3)^{\frac{3}{2} }+c$

Explicação passo a passo:

$\displaystyle\int3x^2\sqrt{1+x^3}~dx \\\\Seja~1+x^3=u~~(diferenciando)\\\\3x^2dx=du\\\\ \displaystyle\int3x^2\sqrt{1+x^3}~dx=\displaystyle\int\sqrt{1+x^3}~3x^2dx=\displaystyle\int\sqrt{u}~du= \displaystyle\int{u^{\frac{1}{2} } ~du=\frac{u^\frac{3}{2} }{\frac{3}{2} } +c=\frac{2}{3}(u^{\frac{3}{2} } +c_1)=$$\frac{2}{3} (1+x^3)^{\frac{3}{2} } +\frac{2}{3}c_1= \frac{2}{3} (1+x^3)^{\frac{3}{2} }+c$