Evaluate: $integral fraction numerator sin left parenthesis straight x minus straight a right parenthesis over denominator sin left parenthesis straight x plus straight a right parenthesis end fraction dx$

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Solution

straight I space equals space integral fraction numerator sin left parenthesis straight x minus straight a right parenthesis over denominator sin left parenthesis straight x plus straight a right parenthesis 1 end fraction dx

Let space left parenthesis straight x plus straight a right parenthesis space equals space straight t space rightwards double arrow space dx space equals space dt therefore space straight I space equals space integral fraction numerator sin left parenthesis straight t minus 2 straight a right parenthesis over denominator sint end fraction dt equals space integral fraction numerator sint space cos 2 straight a minus cost space sin 2 straight a over denominator sin space straight t end fraction dt equals integral left parenthesis cos 2 straight a minus cot space straight t space sin 2 straight a right parenthesis space dt equals left parenthesis cos 2 straight a right parenthesis straight t minus sin 2 straight a space log open vertical bar sin space straight t close vertical bar plus straight C equals cos 2 straight a left parenthesis straight x plus straight a right parenthesis minus sin 2 straight a space log open vertical bar sin left parenthesis straight x plus straight a right parenthesis close vertical bar plus straight C

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