First slide

Methods of integration

Question

The value of $\int \frac{\sin x - \cos x}{\sqrt{\sin 2 x - 1 / 2}} d x$ is

Easy

A

$\log | (\sin x + \cos x) + \sqrt{\sin 2 x} | + C$
B

$- \log | (\sin x + \cos x) + \sqrt{\sin 2 x - 1 / 2} | + C$
C

$- \frac{1}{\sqrt{2}} \log | (\sin x + \cos x) + \sqrt{\sin 2 x - 1 / 2} | + C$
D

none of these

Solution

The given integral can be written as
$\int \frac{\sin x - \cos x}{\sqrt{(\sin x + \cos x)^{2} - 3 / 2}} d x = - \int \frac{d t}{\sqrt{t^{2} - 3 / 2}}$
$(t = \sin x + \cos x)$
$\begin{array}{l} = - \log |t + \sqrt{t^{2} - 3 / 2}| + C \\ = - \log | \sin x + \cos x + \sqrt{\sin 2 x - 1 / 2} | + C \end{array}$

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