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Trigonometric Ratios of Compound Angles

Integrate the...

Question

Integrate the following functions w.r.t. x.

$\int \frac{s i n x}{s i n (x - a)} d x .$

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Solution

$\int \frac{s i n x}{s i n (x - a)} d x = \int \frac{s i n {(x - a) + a}}{s i n (x - a)} d x = \int \frac{s i n (x - a) c o s a + c o s (x - a) s i n a}{s i n (x - a)} d x [∵ s i n (A + B) = s i n A c o s B + c o s A s i n B] = \int \frac{s i n (x - a) c o s a}{s i n (x - a)} d x + \int \frac{c o s (x - a) s i n a}{s i n (x - a)} d x = c o s a \int 1 d x + s i n a \int c o t (x - a) d x = (c o s a) x + s i n a . l o g | s i n (x - a) | + C$

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