Use the substitution $u = 1 - x^3$ to find $\int \frac{x^5}{\sqrt{1 - x^3}} \, \mathrm{d}x$. Hence find $\int \frac{x^8}{\sqrt{1 - x^3}} \, \mathrm{d}x$.

1. Differentiate the substitution to express $\mathrm{d}x$ in terms of $\mathrm{d}u$.

Let $u = 1 - x^3$.
Then $\frac{\mathrm{d}u}{\mathrm{d}x} = -3x^2$.
$\implies x^2 \, \mathrm{d}x = -\frac{1}{3} \, \mathrm{d}u$

2. Rewrite the integral entirely in terms of $u$.

We can split $x^5$ as $x^3 \cdot x^2$:
$\int \frac{x^3 \cdot x^2}{\sqrt{1 - x^3}} \, \mathrm{d}x$

Since $x^3 = 1 - u$, substitute $u$ into the integral:
$\int \frac{1 - u}{\sqrt{u}} \left( -\frac{1}{3} \right) \, \mathrm{d}u$
$= -\frac{1}{3} \int \left( u^{-1/2} - u^{1/2} \right) \, \mathrm{d}u$

3. Integrate the expression with respect to $u$.

$= -\frac{1}{3} \left[ \frac{u^{1/2}}{1/2} - \frac{u^{3/2}}{3/2} \right] + C$
$= -\frac{1}{3} \left( 2u^{1/2} - \frac{2}{3}u^{3/2} \right) + C$
$= -\frac{2}{3}u^{1/2} + \frac{2}{9}u^{3/2} + C$

4. Substitute $x$ back to get the final expression.

Substituting $u = 1 - x^3$ back into the result:
$= -\frac{2}{3}(1 - x^3)^{1/2} + \frac{2}{9}(1 - x^3)^{3/2} + C$

1. Apply the substitution $u = 1 - x^3$ to express the integral entirely in terms of $u$.

Using the substitution introduced in part (a), $u = 1 - x^3$, we differentiate to get $\frac{\mathrm{d}u}{\mathrm{d}x} = -3x^2$, which rearranges to $x^2 \, \mathrm{d}x = -\frac{1}{3} \mathrm{d}u$.
Rearranging the substitution also gives $x^3 = 1 - u$, meaning $x^6 = (1 - u)^2$.
We separate $x^8$ in the integrand into $x^6 \cdot x^2$:
$\int \frac{x^8}{\sqrt{1 - x^3}} \, \mathrm{d}x = \int \frac{x^6}{\sqrt{1 - x^3}} \cdot x^2 \, \mathrm{d}x$
Substituting our expressions in terms of $u$:
$= \int \frac{(1 - u)^2}{\sqrt{u}} \left(-\frac{1}{3}\right) \, \mathrm{d}u$

2. Expand the integrand into a sum of power functions of $u$.

Expand $(1 - u)^2 = 1 - 2u + u^2$ and divide each term by $u^{1/2}$:
$-\frac{1}{3} \int \frac{1 - 2u + u^2}{u^{1/2}} \, \mathrm{d}u$
$= -\frac{1}{3} \int \left( u^{-1/2} - 2u^{1/2} + u^{3/2} \right) \, \mathrm{d}u$

3. Integrate term-by-term using the power rule for integration.

Integrate with respect to $u$, remembering to add the constant of integration $C$:
$= -\frac{1}{3} \left( \frac{u^{1/2}}{1/2} - 2\frac{u^{3/2}}{3/2} + \frac{u^{5/2}}{5/2} \right) + C$
$= -\frac{1}{3} \left( 2u^{1/2} - \frac{4}{3}u^{3/2} + \frac{2}{5}u^{5/2} \right) + C$
$= -\frac{2}{3}u^{1/2} + \frac{4}{9}u^{3/2} - \frac{2}{15}u^{5/2} + C$

4. Substitute $u = 1 - x^3$ back into the expression to obtain the final answer.

Replace all instances of $u$ with $1 - x^3$:
$= -\frac{2}{3}(1 - x^3)^{1/2} + \frac{4}{9}(1 - x^3)^{3/2} - \frac{2}{15}(1 - x^3)^{5/2} + C$