Question

Find the integrals (primitives):
$\rm \displaystyle\int \sqrt{1-\cos 2x} \ dx$

Answer 1

We know that

$cos \: 2x = 1 - 2 {sin}^{2} x \\ \\ so \\ \\ \sqrt{1 - cos \: 2x} = \sqrt{1 - 1 + 2 {sin}^{2} x} \\ \\ = \sqrt{2 {sin}^{2}x } \\ \\ = \sqrt{2} sin \: x$
$\int \sqrt{1-\cos 2x} \ dx \\ \\\int \: \sqrt{2} sin \: x \: dx \\ \\ \sqrt{2} \int \: sin \: x \: dx = \sqrt{2} ( - cos \: x) + C \\ \\ \int \sqrt{1-\cos 2x} \ dx = - \sqrt{2} cos \: x + C$
Hope it helps you.