Question

Usando u = x+7 determine
∫ x (x+7)³ dx

Answer 1

$\displaystyle\int x(x+7)^{3}dx$

Fazendo $u=x+7$, temos $x=u-7$ e $du=dx$. Logo:

$\displaystyle x(x+7)^{3}dx=\int(u-7)u^{3}\,du=\int(u^{4}-7u^{3})du\\\\\\=\dfrac{u^{4+1}}{4+1}-7\dfrac{u^{3+1}}{3+1}+C=\dfrac{u^{5}}{5}-\dfrac{7}{4}u^{4}+C$

Precisamos retornar para x. Para isso, basta usar $u=x+7$:

$\boxed{\boxed{\int x(x+7)^{3}dx=\dfrac{(x+7)^{5}}{5}-\dfrac{7(x+7)^{4}}{4}+C}}$