Solve y=sqrt{1-cosx}dx

$$y = \sqrt { 1 - \cos x } d x$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=\frac{\sqrt{2}y\times \left(2\left(-\cos(x)+1\right)\right)^{-\frac{1}{2}}}{x}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }x=2\pi n_{1}\\d\in \mathrm{C}\text{, }&y=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }x=2\pi n_{1}\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{1-\cos(x)}dx=y$$

The equation is in standard form.

$$x\sqrt{-\cos(x)+1}d=y$$

Divide both sides by \sqrt{1-\cos(x)}x.

$$\frac{x\sqrt{-\cos(x)+1}d}{x\sqrt{-\cos(x)+1}}=\frac{y}{x\sqrt{-\cos(x)+1}}$$

Dividing by \sqrt{1-\cos(x)}x undoes the multiplication by \sqrt{1-\cos(x)}x.

$$d=\frac{y}{x\sqrt{-\cos(x)+1}}$$

Divide y by \sqrt{1-\cos(x)}x.

$$d=\frac{y\left(-\cos(x)+1\right)^{-\frac{1}{2}}}{x}$$

Solve for d

$$\left\{\begin{matrix}d=\frac{y}{x\sqrt{-\cos(x)+1}}\text{, }&\nexists n_{1}\in \mathrm{Z}\text{ : }x=2\pi n_{1}\\d\in \mathrm{R}\text{, }&y=0\text{ and }\exists n_{1}\in \mathrm{Z}\text{ : }x=2\pi n_{1}\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{1-\cos(x)}dx=y$$

The equation is in standard form.

$$x\sqrt{-\cos(x)+1}d=y$$

Divide both sides by \sqrt{1-\cos(x)}x.

$$\frac{x\sqrt{-\cos(x)+1}d}{x\sqrt{-\cos(x)+1}}=\frac{y}{x\sqrt{-\cos(x)+1}}$$

Dividing by \sqrt{1-\cos(x)}x undoes the multiplication by \sqrt{1-\cos(x)}x.

$$d=\frac{y}{x\sqrt{-\cos(x)+1}}$$