Official solution
Let \[I = \sqrt 2 \int {{{\sin \,xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}} \]
Put \[x - {\pi \over 4} = t\]
\[ \Rightarrow dx = dt\]
\[ \Rightarrow I = \sqrt 2 \int {{{\sin \left( {t + {\pi \over 4}} \right)} \over {\sin \,t}}} dt\]
\[\,\,\,\,\,\,\,\,\,\,\,\, = {{\sqrt 2 } \over {\sqrt 2 }}\int {\left( {{{\sin t + \cos t} \over {\sin t}}} \right)} \,\,dt\]
\[ \Rightarrow I = \int {\left( {1 + \cot \,t} \right)} dt\]
\[\,\,\,\,\,\,\,\,\,\,\,\, = t + \log \left| {\sin t} \right| + {c_1}\]
\[ = x - {\pi \over 4} + \log \left| {\sin \left( {x - {\pi \over 4}} \right)} \right| + {c_1}\]
\[ = x + \log \left| {\sin \left( {x - {\pi \over 4}} \right)} \right| + c\]
\[\left( \, \right.\] where \[{c = {c_1} - {\pi \over 4}}\] \[\left. \, \right)\]