Answer
$-\frac{(4-y^3)^{5/3}}{5} + C$
Work Step by Step
Evaluate the Integral using substitution: $\int y^2(4-y^3)^{2/3}dy$
Substitution Rule: $\int f(g(x))gā(x)dx = \int f(u)du$
$u= 4-y^3$
$du =-3y^2$
Since $du$ in the expression is equal to $y^2$ it must be multiplied by $-\frac{1}{3}$
Solve the integral in terms of $u$:
$\int (u)^{2/3}(-\frac{1}{3})du$
$-\frac{1}{3}\int u^{2/3}du $
$-\frac{1}{3} \frac{3u^{5/3}}{5} + C$
$-\frac{u^{5/3}}{5} + C$
Substitute for $u$:
$-\frac{(4-y^3)^{5/3}}{5} + C$