First slide

Methods of integration

Question

$\int \frac{\sin x}{\sin (x - a)} d x =$

Moderate

A

$x \sin a + \cos a \log |\sin (x - a)| + C$
B

$\cos x + \sin x \log |\sin (x - a)| + C$
C

$(x - a) \cos a + \sin a \log |\sin (x - a)| + C$
D

$x \cos a + \sin a \log |\sin (x - a)| + C$

Solution

$I = \int \frac{\sin x}{\sin (x - a)} d x$
$\begin{array}{l} = \int \frac{\sin (x - a + a)}{\sin (x - a)} d x \\ = \int \frac{\sin (x - a) \cos a + \cos (x - a) \sin a}{\sin (x - a)} d x \\ = \int [\cos a + \sin a \cot (x - a)] d x \\ = \cos a \int d x + \sin a \int \cot (x - a) d x \\ = x \cos a + \sin a \log | \sin (x - a) | + C \end{array}$

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