Question

integral de x (2x^2+3)^10

Answer 1

$I=\displaystyle\int{x\,(2x^{2}+3)^{10}\,dx}\\\\\\ =\dfrac{1}{4}\displaystyle\int{4x\,(2x^{2}+3)^{10}\,dx}\\\\\\ =\dfrac{1}{4}\displaystyle\int{(2x^{2}+3)^{10}\cdot 4x\,dx}~~~~~~\mathbf{(i)}$


Fazendo a seguinte substituição:

$2x^{2}+3=u~~\Rightarrow~~4x\,dx=du$


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

$=\displaystyle\dfrac{1}{4}\displaystyle\int{u^{10}\,du}\\\\\\ =\dfrac{1}{4}\cdot \dfrac{u^{10+1}}{10+1}+C\\\\\\ =\dfrac{1}{4}\cdot \dfrac{u^{11}}{11}+C\\\\\\ =\dfrac{u^{11}}{44}+C\\\\\\ =\dfrac{(2x^{2}+3)^{11}}{44}+C\\\\\\\\ \therefore~~\boxed{\begin{array}{c} \displaystyle\int{x\,(2x^{2}+3)^{10}\,dx}=\dfrac{(2x^{2}+3)^{11}}{44}+C \end{array}}$