The value of $$\sqrt 2 \int {{{\sin xdx} \over {\sin \left( {x - {\pi \over 4}} \right)}}} $$ is
A
$$\,x + \log \,\left| {\,\cos \left( {x - {\pi \over 4}} \right)\,} \right| + c$$
B
$$\,x - \log \,\left| {\,\sin \left( {x - {\pi \over 4}} \right)\,} \right| + c$$
C
$$\,x + \log \,\left| {\,\sin \left( {x - {\pi \over 4}} \right)\,} \right| + c$$
D
$$\,x - \log \,\left| {\,\cos \left( {x - {\pi \over 4}} \right)\,} \right| + c$$
Explanation
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 $${c = {c_1} - {\pi \over 4}}$$ $$\left. \, \right)$$