# How do you integrate int (9+x^2)/sqrt(4 - x^2)dx using trigonometric substitution?

##### 1 Answer
Apr 2, 2018

$11 {\sin}^{- 1} \left(\frac{x}{2}\right) + 4 \sin \left(2 {\sin}^{- 1} \left(\frac{x}{2}\right)\right)$

#### Explanation:

$\int \frac{9 + {x}^{2}}{\sqrt{4 - {x}^{2}}}$
Sub x=2 sin $\theta$
$\frac{\mathrm{dx}}{d} \left(\theta\right)$=2cos$\theta$
$\int {\left(9 + 4 \left(\sin \theta\right)\right)}^{2} / \sqrt{4 {\left(\cos \theta\right)}^{2}}$ x 2cos$\theta$
=$\int 9 + 4 {\left(\sin \theta\right)}^{2}$
= $\int 9 + 4 \times \frac{1}{2} \left(1 - \cos 2 \theta\right)$

--> $\cos 2 \theta = {\left(\cos \theta\right)}^{2} - {\left(\sin \theta\right)}^{2}$
--> $\cos 2 \theta = 1 - 2 {\left(\sin \theta\right)}^{2}$
--> ${\left(\sin \theta\right)}^{2} = \frac{1}{2} \left(1 - \cos 2 \theta\right)$

= $\int 9 + 2 - 2 \cos 2 \theta$
= $\int 11 - 2 \cos 2 \theta$
= $11 \theta + 4 \sin 2 \theta$
= $11 {\sin}^{- 1} \left(\frac{x}{2}\right) + 4 \sin \left(2 {\sin}^{- 1} \left(\frac{x}{2}\right)\right)$