First slide

Methods of integration

Question

The value of $\sqrt{2} \int \frac{\sin x}{\sin (x - \frac{π}{4})} d x,$ is

Moderate

A

$x + \log |\sin (x - \frac{π}{4})| + C$
B

$x - \log |\cos (x - \frac{π}{4})| + C$
C

$x + \log |\cos (x - \frac{π}{4})| + C$
D

$x - \log |\sin (x - \frac{π}{4})| + C$

Solution

We have,
$\begin{array}{l} \sqrt{2} \int \frac{\sin x}{\sin (x - \frac{π}{4})} d x \\ = \sqrt{2} \int \frac{\sin \{(x - \frac{π}{4}) + \frac{π}{4}\}}{\sin (x - \frac{π}{4})} d x \\ = \sqrt{2} \int \frac{\sin (x - \frac{π}{4}) \cos \frac{π}{4} + \cos (x - \frac{π}{4}) \sin \frac{π}{4}}{\sin (x - \frac{π}{4})} d x \\ = \int 1 \cdot d x + \int \cot (x - \frac{π}{4}) d x = x + \log |\sin (x - \frac{π}{4})| + C \end{array}$

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