# What is the integral of int cosx/sqrt(1-2sinx)?

Mar 31, 2018

$- \sqrt{1 - 2 \sin \left(x\right)} + c$

#### Explanation:

Start by substituting $u = 1 - 2 \sin \left(x\right)$ and $\mathrm{du} = - 2 \cos \left(x\right) \mathrm{dx}$
$\int \cos \frac{x}{\sqrt{1 - 2 \sin x}} \mathrm{dx} = \int \frac{- \frac{1}{2}}{\sqrt{u}} \mathrm{du} = - \frac{1}{2} \int \frac{1}{\sqrt{u}} \mathrm{du}$
$\int \frac{1}{\sqrt{u}} \mathrm{du} = \int {u}^{- \frac{1}{2}} \mathrm{du} = 2 {u}^{\frac{1}{2}} + c$

All in all:
$- \frac{1}{2} \cdot 2 {u}^{\frac{1}{2}} + c = - \sqrt{u} + c$

Substitute back

$- \sqrt{u} + c = - \sqrt{1 - 2 \sin \left(x\right)} + c$