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Question Number 180398 by a.lgnaoui last updated on 12/Nov/22

∫(1/(sin (x−a)sin (x−b)))dx=?

$$\int\frac{\mathrm{1}}{\mathrm{sin}\:\left(\mathrm{x}−\mathrm{a}\right)\mathrm{sin}\:\left(\mathrm{x}−\mathrm{b}\right)}\mathrm{dx}=? \\ $$$$ \\ $$

Commented by Spillover last updated on 12/Nov/22

https://youtu.be/J0cicATsvxM

Answered by MJS_new last updated on 12/Nov/22

∫(dx/(sin (x−a) sin (x−b)))=  =(1/(sin (a−b)))∫(cot (x−a) −cot (x−b))dx=  =(1/(sin (a−b)))(ln ∣sin (x−a)∣ −ln ∣sin (x−b)∣)=  =(1/(sin (a−b)))ln ∣((sin (x−a))/(sin (x−b)))∣ +C

$$\int\frac{{dx}}{\mathrm{sin}\:\left({x}−{a}\right)\:\mathrm{sin}\:\left({x}−{b}\right)}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{sin}\:\left({a}−{b}\right)}\int\left(\mathrm{cot}\:\left({x}−{a}\right)\:−\mathrm{cot}\:\left({x}−{b}\right)\right){dx}= \\ $$$$=\frac{\mathrm{1}}{\mathrm{sin}\:\left({a}−{b}\right)}\left(\mathrm{ln}\:\mid\mathrm{sin}\:\left({x}−{a}\right)\mid\:−\mathrm{ln}\:\mid\mathrm{sin}\:\left({x}−{b}\right)\mid\right)= \\ $$$$=\frac{\mathrm{1}}{\mathrm{sin}\:\left({a}−{b}\right)}\mathrm{ln}\:\mid\frac{\mathrm{sin}\:\left({x}−{a}\right)}{\mathrm{sin}\:\left({x}−{b}\right)}\mid\:+{C} \\ $$

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