Question

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

Answer 1

We know that

${sin}^{2} \frac{x}{2} + {cos}^{2} \frac{x}{2} = 1 \\ \\ sin \: x = 2sin \: \frac{x}{2} \: cos \: \frac{x}{2} \\ \\ put \: these \: values \: in \: the \: integration \\ \\ \sqrt{1 - sin \: x} = \sqrt{{sin}^{2} \frac{x}{2} + {cos}^{2} \frac{x}{2} -2sin \: \frac{x}{2} \: cos \: \frac{x}{2} } \\ \\ = \sqrt{ {(sin \: \frac{x}{2} - cos \: \frac{x}{2}) }^{2} } \\ \\ = sin \: \frac{x}{2} - cos \: \frac{x}{2} \\ \\ now \: integrate$
$\int \: sin \: \frac{x}{2} - cos \: \frac{x}{2} \\ \\ = 2( - cos \: \frac{x}{2}) - 2 \: sin \: \frac{x}{2} + C \\ \\ = - 2 cos \: \frac{x}{2} - 2 \: sin \: \frac{x}{2} + C \\ \\ \sqrt{1 - sin \: x} = - 2(cos \: \frac{x}{2} + sin \: \frac{x}{2} ) + C$
Hope it helps you.