Question

Qual a integral de 2x√(x-2)dx

Answer 1

$\displaystyle\mathsf{\int2x\sqrt{x-2} \: dx}$

$\mathsf{t=\sqrt{x-2}\to~x-2={t}^{2}} \\\mathsf{x = 2 + {t}^{2} \to \: dx = 2t \: dt}$

$\displaystyle\mathsf{\int2x\sqrt{x-2} \: dx } \\ = \mathsf{ \int2.( {t}^{2} + 2).t.2tdt} \\ \mathsf{4\int( {t}^{4} + 2{t}^{2})dt = \dfrac{4}{5}{t}^{5} + \dfrac{8}{3}{t}^{3} + k}$

$\displaystyle\mathsf{\int2x\sqrt{x-2} \: dx}\\ = \mathsf{\dfrac{4}{5}{(\sqrt{x-2})}^{5}+\dfrac{8}{3}{(\sqrt{x-2})}^{3}+k}$