Question

calcule a integral indefinida: $\int\limits\ \sqrt{x+1}* {x} \, dx$

Answer 1

$\displaystyle\mathsf{\int\sqrt{x+1}~x \: dx}$

faça

$\mathsf{u=\sqrt{x+1}\to~x+1=u^2}\\\mathsf{x=u^2-1\to~dx=2u \: du}$

Substituindo

$\displaystyle\mathsf{\int\sqrt{x+1}x~dx=\int u.(u^2-1).2u~du} \\ \mathsf{=2\int(u^4-u^2)du = \dfrac{2}{5}{u}^{5} -\dfrac{2}{3} {u}^{3} + k}$

Voltando

$\displaystyle\mathsf{\int\sqrt{x+1} \: x~dx} \\\mathsf{=\dfrac{2}{5}(\sqrt{x+1})^5-\dfrac{2}{3}(\sqrt{x+1})^3+k}$