Could someone please help evaluate the integral $\displaystyle \int {x}^{5}\sqrt{1-{x}^{3} }$
Let $\displaystyle u = 1 - x^3$
$\displaystyle du = - 3x^2 dx$
So we have,
$\displaystyle -\frac{1}{3}\int {x}^{3}\sqrt{u} \ du$
But we said $\displaystyle u = 1 - x^3$, so rearranging we get $\displaystyle x^3 = (1 - u)$
So sub that in:
= $\displaystyle -\frac{1}{3}\int(1 - u)\sqrt{u} \ du$
= $\displaystyle -\frac{1}{3}\int\sqrt{u} - u^{3/2} \ du$
Now it's easy. Just remember to sub in the value of u after you've done the integration.
Alternatively, we can consider it as a binomial differential
$\displaystyle I = \int x^m(a+b x^n)^p\;dx\quad (m,n,p\in\mathbb{Q}) $
In our case
$\displaystyle \dfrac{m+1}{n}\in\mathbb{Z}$
So, with the substitution
$\displaystyle a+bx^n=t^s\;,\quad s=\textrm{denom}(p)$
we obtain a rational integral on t.