# How do you integrate intx^3sqrt(16 - x^2) dx?

Jun 24, 2015

I would let:
$x = 4 \sin \theta$
$\mathrm{dx} = 4 \cos \theta d \theta$
$\sqrt{16 - {x}^{2}} = 4 \cos \theta$
${x}^{3} = 64 {\sin}^{3} \theta$

$\implies 1024 \int {\sin}^{3} \theta {\cos}^{2} \theta d \theta$

$= 1024 \int {\sin}^{2} \theta \sin \theta {\cos}^{2} \theta d \theta$

$= 1024 \int \left(1 - {\cos}^{2} \theta\right) {\cos}^{2} \theta \sin \theta d \theta$

At this point you can let:
$u = \cos \theta$
$\mathrm{du} = - \sin \theta d \theta$

$\implies 1024 \int \left({\cos}^{2} \theta - 1\right) {\cos}^{2} \theta \left(- \sin \theta\right) d \theta$

$= 1024 \int \left({u}^{2} - 1\right) {u}^{2} \mathrm{du}$

$= 1024 \int {u}^{4} - {u}^{2} \mathrm{du}$

$= 1024 \left({u}^{5} / 5 - {u}^{3} / 3\right)$

$= 1024 \left({\cos}^{5} \frac{\theta}{5} - {\cos}^{3} \frac{\theta}{3}\right)$

Since $\cos \theta = \frac{\sqrt{16 - {x}^{2}}}{4}$:

$= 1024 \left({\left(\frac{\sqrt{16 - {x}^{2}}}{4}\right)}^{5} / 5 - {\left(\frac{\sqrt{16 - {x}^{2}}}{4}\right)}^{3} / 3\right)$

$= 1024 \left(\frac{{\left(16 - {x}^{2}\right)}^{\frac{5}{2}} / 1024}{5} - \frac{{\left(16 - {x}^{2}\right)}^{\frac{3}{2}} / 64}{3}\right)$

$= 1024 \left({\left(16 - {x}^{2}\right)}^{\frac{5}{2}} / \left(5 \cdot 1024\right) - {\left(16 - {x}^{2}\right)}^{\frac{3}{2}} / \left(3 \cdot 64\right)\right)$

$= \cancel{1024} \left({\left(16 - {x}^{2}\right)}^{\frac{5}{2}} / \left(5 \cdot \cancel{1024}\right) - {\left(16 - {x}^{2}\right)}^{\frac{3}{2}} / \left(\frac{3}{16} \cdot \cancel{1024}\right)\right)$

$= \textcolor{g r e e n}{\frac{1}{5} {\left(16 - {x}^{2}\right)}^{\text{5/2") - 16/3(16-x^2)^("3/2}} + C}$

Normally here would be okay, but we can go a bit further.

$= {\left(16 - {x}^{2}\right)}^{\frac{3}{2}} \left(\frac{1}{5} \left(16 - {x}^{2}\right) - \frac{16}{3}\right)$

$= {\left(16 - {x}^{2}\right)}^{\frac{3}{2}} \left(\frac{3}{15} \left(16 - {x}^{2}\right) - \frac{80}{15}\right)$

$= \frac{1}{15} {\left(16 - {x}^{2}\right)}^{\frac{3}{2}} \left(3 \left(16 - {x}^{2}\right) - 80\right)$

$= \frac{1}{15} {\left(16 - {x}^{2}\right)}^{\frac{3}{2}} \left(48 - 3 {x}^{2} - 80\right)$

$= \frac{1}{15} {\left(16 - {x}^{2}\right)}^{\frac{3}{2}} \left(- 32 - 3 {x}^{2}\right)$

$= \textcolor{b l u e}{- \frac{1}{15} {\left(16 - {x}^{2}\right)}^{\text{3/2}} \left(3 {x}^{2} + 32\right) + C}$