If∫$$1sin$$⁡(x$$−$$a)$$sin$$⁡(x$$−$$b)dx$$ $$=A$$ $$logsin(x-a)$$sin$$$$(x-b)+c$$

Question

If∫$$1sin$$⁡(x$$−$$a)$$sin$$⁡(x$$−$$b)dx$$ $$=A$$ $$logsin(x-a)$$sin$$$$(x-b)+c$$

Infinity Learn · Accepted Answer

∫$$1sin$$⁡(x$$−$$a)$$sin$$⁡(x$$−$$b)dx=1sin$$⁡$$(a$$−$$b)∫$$sin$$⁡$${(x$$−$$b)$$−(x$$−$$a)}$$$$sin$$⁡$$(x$$−$$a)$$sin$$⁡(x$$−$$b)dx$$ $$=1sin$$⁡$$(a$$−$$b)∫$$sin$$⁡(x$$−$$b)$$cos$$⁡(x$$−$$a)−$$cos$$⁡(x$$−$$b)$$sin$$⁡(x$$−$$a)$$sin$$⁡(x$$−$$a)$$sin$$⁡(x$$−$$b)dx=1sin$$⁡$$(a$$−$$b)∫$${$$$$cot$$⁡(x$$−$$a)−$$cot$$⁡(x$$−$$b)}dx=1sin$$⁡$$(a$$−$$b){log$$⁡|$$sin$$⁡$$(x$$−$$a)|−log⁡|$$sin$$⁡(x$$−$$b)|$$}+c=cosec$$⁡(a$$−$$b)log$$⁡$$sin$$⁡$$(x$$−$$a)$$sin$$⁡$$(x$$−$$b)+c$$

$If \int \frac{1}{\sin (x - a) \sin (x - b)} d x = A \log |\frac{\sin (x - a)}{\sin (x - b)}| + c$

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If∫1sin⁡(x−a)sin⁡(x−b)dx =A logsin(x-a)sin(x-b)+c

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$If \int \frac{1}{\sin (x - a) \sin (x - b)} d x = A \log |\frac{\sin (x - a)}{\sin (x - b)}| + c$