Integral of $$$\sqrt{4 - 4 \sin^{2}{\left(x \right)}}$$$

Find $$$\int \sqrt{4 - 4 \sin^{2}{\left(x \right)}}\, dx$$$.

Simplify the integrand:

$${\color{red}{\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x}}} = {\color{red}{\int{2 \sqrt{1 - \sin^{2}{\left(x \right)}} d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=2$$$ and $$$f{\left(x \right)} = \sqrt{1 - \sin^{2}{\left(x \right)}}$$$:

$${\color{red}{\int{2 \sqrt{1 - \sin^{2}{\left(x \right)}} d x}}} = {\color{red}{\left(2 \int{\sqrt{1 - \sin^{2}{\left(x \right)}} d x}\right)}}$$

This integral (Incomplete Elliptic Integral of the Second Kind) does not have a closed form:

$$2 {\color{red}{\int{\sqrt{1 - \sin^{2}{\left(x \right)}} d x}}} = 2 {\color{red}{E\left(x\middle| 1\right)}}$$

Therefore,

$$\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x} = 2 E\left(x\middle| 1\right)$$

Add the constant of integration:

$$\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x} = 2 E\left(x\middle| 1\right)+C$$

$$$\int \sqrt{4 - 4 \sin^{2}{\left(x \right)}}\, dx = 2 E\left(x\middle| 1\right) + C$$$A