The correct option is D cos(b a)x + sin (b a) log | sin ( x b )| +c ∫sin[(x b) (a b)]sin(x b)dx ∫ ( sin(x b)cos(a b) sin(a b)cos(x b)sin(x b) ...

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Trigonometric Ratios for Sum of Two Angles

∫sinx-a/sinx-...

Question

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

(x - a) c o s (b - a) + l o g | s i n (x - b) | + c

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(x - b) c o s (a + b) + l o g | s i n (x - a) | + c

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(x - b) c o s (b - a) + l o g | s i n (x - b) | + c

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c o s (b - a) x + s i n (b - a) l o g | s i n (x - b) | + c

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Solution

The correct option is D $c o s (b - a) x + s i n (b - a) l o g | s i n (x - b) | + c$
$\int \frac{s i n [(x - b) - (a - b)]}{s i n (x - b)} d x$

$\int (\frac{s i n (x - b) c o s (a - b) - s i n (a - b) c o s (x - b)}{s i n (x - b)}) d x$

$= \int c o s (a - b) d x - s i n (a - b) \int c o t (x - b) d y$

$= c o s (a - b) \cdot x + s i n (b - a) l o g | s i n (x - b) | + c$

$= c o s (b - a) x + s i n (b - a) l o g | s i n (x - b) | + c$

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