Answer:
[tex] \sin \theta - \cos \theta + c[/tex]
Step-by-step explanation:
[tex] \int \sqrt{1 + \sin 2 \theta} \: d \theta \\ formulae \: to \: be \: ued \\ { \sin}^{2} \theta + { \cos}^{2} \theta = 1 \\ \sin2 \theta = 2 \sin \theta \: \cos \theta \\ \\ \therefore \: \int \sqrt{1 + \sin 2 \theta} \: d \theta \\ = \int \sqrt{ { \sin}^{2} \theta + { \cos}^{2} \theta + 2 \sin \theta \: \cos \theta} \: d \theta \\ = \int \sqrt{ {(\sin \theta \: + \cos \theta )}^{2} } \: d \theta \\ = \int( \sin \theta + \cos \theta) \: d \theta \\ = \int \sin \theta \: d \theta + \int \cos \theta \: d \theta \\ = - \cos \theta + \sin \theta + c \\ \huge \red{ \boxed{ = \sin \theta - \cos \theta + c}}[/tex]
