[tex] \sin \theta = \frac{x}{ \sqrt{2} } \: \to \: x = \sqrt{2} . \sin \theta \\ \\ x = \sqrt{2} . \sin \theta \: \: \to \: \frac{dx}{d \theta} = \sqrt{2} . \cos \theta \\ \\ dx = \sqrt{2} . \cos \theta d \theta[/tex]

[tex] \int \frac{1}{( \sqrt{2}. \sin \theta)^{2} . \sqrt{2 - ( \sqrt{2}. \sin \theta) {}^{2} } } . \sqrt{2} . \cos \theta \: d \theta \\ \\ \int \frac{ \sqrt{2}. \cos \theta d \theta }{2 \sin^{2} \theta. \sqrt{2 -2 \sin^{2} \theta } } \: \to \: \int \frac{ \sqrt{2}. \cos \theta d \theta }{2 \sin^{2} \theta. \sqrt{2.(1 - \sin^{2} \theta) } } \\ \\ \int \frac{ \sqrt{2}. \cos \theta d \theta }{2 \sin^{2} \theta. \sqrt{2.(\cos^{2} \theta) } } \: \to \: \int \frac{ \sqrt{2} . \cos \theta d \theta}{2 \sin^{2} \theta \sqrt{2}. \cos \theta } \\ \\ \int \frac{1}{2 \sin {}^{2} \theta} d \theta \: \to \: \frac{1}{2} \int \csc {}^{2} ( \theta) \: d \theta \\ \\ \boxed{ \frac{1}{2} .( - \cot(\theta)) + c}[/tex]

[tex] \sin \theta = \frac{x}{ \sqrt{2} } \: \: e \: \: \cos \theta = \frac{ \sqrt{2 - x {}^{2} } }{ \sqrt{2} } \\ \\ \tg \theta = \frac{ \frac{x}{ \sqrt{2} } }{ \frac{ \sqrt{2 - x {}^{2} } }{ \sqrt{2} } } \: \to \: \tg \theta = \frac{x}{ \sqrt{2 - x {}^{2} } } \\ \\ \cot \theta = \frac{ \sqrt{2 - x {}^{2} } }{x} [/tex]

[tex] \frac{1}{2} . ( - \cot( \theta)) + c \: \: \to \: \: - \frac{1}{2} . \left( \frac{ \sqrt{2 - x {}^{2} } }{x} \right) + c \\ \\ \boxed{\boxed{\boxed{ - \frac{ \sqrt{2 - x {}^{2} } }{2x} + c} }}[/tex]