First slide

Methods of integration

Question

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

Easy

A

$x + \log | \cos (x - π / 4) | + C$
B

$x - \log | \sin (x - π / 4) | + C$
C

$x + \log | \sin (x - π / 4) | + C$
D

$x - \log | \cos (x - π / 4) | + C$

Solution

Put $x - π / 4 = t$
$\begin{aligned} I = \sqrt{2} \int \frac{\sin (π / 4 + t)}{\sin t} d t = \int (\cot t + 1) d t \\ = t + \log | \sin t | + C^{'} \\ = x + \log | \sin (x - π / 4) | + C, C = C^{'} + π / 4 \end{aligned}$

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