Question

Calcule a integral abaixo:
$\int\ { x^{2} ( x^{3}-1) ^{10} } \, dx$

Answer 1

É dada a integral:

$I=\displaystyle \int x^2(x^3-1)^{10}\,dx$

Seja $y=x^3-1$. Então: $dy=3x^2dx\Longrightarrow x^2dx=\dfrac{dy}{3}$. Substituindo na integral:

$\displaystyle I=\int x^2(x^3-1)^{10}\,dx\\\\ I=\int y^{10}\dfrac{dy}{3}}\\\\ I=\frac{1}{3}\int y^{10}\,dy\\\\ I=\frac{1}{3}\cdot\dfrac{y^{10+1}}{10+1}+C\\\\ I=\frac{1}{3}\cdot\dfrac{y^{11}}{11}+C\\\\ I=\dfrac{y^{11}}{33}+C$

Voltando à expressão em x:

$I=\dfrac{y^{11}}{33}+C,~~~~y=x^3-1\\\\ I=\dfrac{(x^3-1)^{11}}{33}+C\\\\ \boxed{\int x^2(x^3-1)^{10}\,dx=\dfrac{(x^3-1)^{11}}{33}+C}$