$\int \frac{\cos x}{\sin (x - \frac{π}{4})} d x =$

Moderate

A

$\frac{1}{\sqrt{2}} \log (\sin (x - \frac{π}{4})) - \frac{x - \frac{π}{4}}{\sqrt{2}}$
B

$\log (\cos (x - \frac{π}{4})) + \frac{x - \frac{π}{4}}{\sqrt{2}} + c$
C

$\frac{1}{\sqrt{2}} \log (\sin (x + \frac{π}{4})) + \frac{x + \frac{π}{4}}{\sqrt{2}} + c$
D

$\frac{1}{\sqrt{2}} \log (\cos (x + \frac{π}{4})) + \frac{x + \frac{π}{4}}{\sqrt{2}} + c$

Solution

$x - \frac{π}{4} = t \Rightarrow d x = d t$
$\begin{array}{l} = \int \frac{\cos (x)}{\sin (x - \frac{π}{4})} d x = \int \frac{\cos (t + \frac{π}{4})}{\sin t} d t \\ = \int \frac{\cos t \frac{1}{\sqrt{2}} - \sin t \frac{1}{\sqrt{2}}}{\sin t} d t \\ = \int \frac{1}{\sqrt{2}} \cot t - \frac{1}{\sqrt{2}} d t \\ = \frac{1}{\sqrt{2}} \log (\sin t) - \frac{t}{\sqrt{2}} + c \\ = \frac{1}{\sqrt{2}} [\log \sin (x - \frac{π}{4})] - \frac{x - \frac{π}{4}}{\sqrt{2}} + c \end{array}$

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