First slide

Methods of integration

Question

The value of $\sqrt{2} \int \frac{\sin xdx}{\sin (x - \frac{π}{4})}$ is equal to

Moderate

A

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

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

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

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

Solution

Let $I = \sqrt{2} \int \frac{\sin x}{\sin (x - \frac{π}{4})} dx$
Put, $x = \frac{π}{4} = t \Rightarrow dx = dt$
$\begin{array}{l} ∴ I = \sqrt{2} \int \frac{\sin (\frac{π}{4} + t) dt}{\sin t} \\ = \sqrt{2} \int [\frac{1}{\sqrt{2}} \cot t + \frac{1}{\sqrt{2}}] dt \\ = \log | \sin t | + t + C \\ = x + \log |\sin (x - \frac{π}{4})| + C \end{array}$

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