Question

integral de x/(x^2+5)^3 dx

Answer 1

$I=\displaystyle\int{\dfrac{x}{(x^{2}+5)^{3}}\,dx}\\\\\\ =\dfrac{1}{2}\int{\dfrac{2x}{(x^{2}+5)^{3}}\,dx}\\\\\\ =\dfrac{1}{2}\int{\dfrac{1}{(x^{2}+5)^{3}}\cdot 2x\,dx}~~~~~~\mathbf{(i)}$


Fazendo a seguinte substituição:

$x^{2}+5=u~~\Rightarrow~~2x\,dx=du$


Substituindo em $\mathbf{(i)},$ a integral fica

$=\dfrac{1}{2}\displaystyle\int{\dfrac{1}{u^{3}}\,du}\\\\\\ =\dfrac{1}{2}\int{u^{-3}\,du}\\\\\\ =\dfrac{1}{2}\cdot \dfrac{u^{-3+1}}{-3+1}+C\\\\\\ =\dfrac{1}{2}\cdot \dfrac{u^{-2}}{-2}+C\\\\\\ =-\,\dfrac{1}{4}\,u^{-2}+C\\\\\\ =-\,\dfrac{1}{4u^{2}}+C\\\\\\ =-\,\dfrac{1}{4\,(x^{2}+5)^{2}}+C\\\\\\\\ \therefore~~\boxed{\begin{array}{c} \displaystyle\int{\dfrac{x}{(x^{2}+5)^{3}}\,dx}=-\,\dfrac{1}{4\,(x^{2}+5)^{2}}+C \end{array}}$