# How do you integrate int x^3/sqrt(144-x^2)dx using trigonometric substitution?

Sep 7, 2016

$- \frac{1}{3} \sqrt{144 - {x}^{2}} \left({x}^{2} + 288\right) + C$

#### Explanation:

Use the substitution $x = 12 \sin \theta$. From here we see that $\mathrm{dx} = 12 \cos \theta d \theta$. Substituting:

$\int {x}^{3} / \sqrt{144 - {x}^{2}} \mathrm{dx} = \int \frac{1728 {\sin}^{3} \theta \left(12 \cos \theta\right)}{\sqrt{144 - 144 {\sin}^{2} \theta}} d \theta$

Factoring $\sqrt{\frac{1}{144}} = \frac{1}{12}$ from the square root:

$= \int \frac{1728 {\sin}^{3} \theta \left(12 \cos \theta\right)}{12 \sqrt{1 - {\sin}^{2} \theta}} d \theta$

Canceling the $12$s and recalling that since ${\sin}^{2} \theta + {\cos}^{2} \theta = 1$, we know that $\cos \theta = \sqrt{1 - {\sin}^{2} \theta}$:

$= 1728 \int \frac{{\sin}^{3} \theta \cos \theta}{\cos} \theta d \theta = 1728 \int {\sin}^{3} \theta d \theta$

We will use here ${\sin}^{2} \theta = 1 - {\cos}^{2} \theta$:

$= 1728 \int {\sin}^{2} \theta \sin \theta d \theta = 1728 \int \left(1 - {\cos}^{2} \theta\right) \sin \theta d \theta$

Now, using the substitution $u = \cos \theta$, so that $\mathrm{du} = - \sin \theta d \theta$:

$= - 1728 \int \left(1 - {u}^{2}\right) \mathrm{du} = 1728 \int {u}^{2} \mathrm{du} - 1728 \int \mathrm{du}$

Integrating using the power rule for integration:

$= 1728 \left({u}^{3} / 3\right) - 1728 u = 576 {u}^{3} - 1728 u$

Reverse substituting with $u = \cos \theta$:

$= 576 {\cos}^{3} \theta - 1728 \cos \theta$

Because our substitution is $\sin \theta = \frac{x}{12}$, we should write this in terms of sine functions. That is, $\cos \theta = \sqrt{1 - {\sin}^{2} \theta}$ and ${\cos}^{3} \theta = {\left(1 - {\sin}^{2} \theta\right)}^{\frac{3}{2}}$:

$= 576 {\left(1 - {\sin}^{2} \theta\right)}^{\frac{3}{2}} - 1728 \sqrt{1 - {\sin}^{2} \theta}$

Factoring:

$= \sqrt{1 - {\sin}^{2} \theta} \left(576 \left(1 - {\sin}^{2} \theta\right) - 1728\right)$

Now using $\sin \theta = \frac{x}{12}$:

$= \sqrt{1 - {x}^{2} / 144} \left(576 \left(1 - {x}^{2} / 144\right) - 1728\right)$

$= \sqrt{\frac{144 - {x}^{2}}{144}} \left(576 - 4 {x}^{2} - 1728\right)$

Factoring $\sqrt{\frac{1}{144}} = \frac{1}{12}$ from the square root and $- 4$ from the parentheses:

$= - \frac{1}{3} \sqrt{144 - {x}^{2}} \left({x}^{2} + 288\right) + C$