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Question Number 177241 by peter frank last updated on 02/Oct/22

                ∫  ((sin x)/(sin (x−a)))dx

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

Answered by cortano1 last updated on 02/Oct/22

 ⇒x−a=y ; ⇒dx=dy  C=∫ ((sin (a+y))/(sin y)) dy=∫( cos a+sin a ((cos y)/(sin y)))dy  C= (x−a)cos a+sin a ∫ ((d(sin y))/(sin y))   C=(x−a)cos a+sin a ln ∣sin (x−a)∣ +k

$$\:\Rightarrow\mathrm{x}−\mathrm{a}=\mathrm{y}\:;\:\Rightarrow\mathrm{dx}=\mathrm{dy} \\ $$$$\mathrm{C}=\int\:\frac{\mathrm{sin}\:\left(\mathrm{a}+\mathrm{y}\right)}{\mathrm{sin}\:\mathrm{y}}\:\mathrm{dy}=\int\left(\:\mathrm{cos}\:\mathrm{a}+\mathrm{sin}\:\mathrm{a}\:\frac{\mathrm{cos}\:\mathrm{y}}{\mathrm{sin}\:\mathrm{y}}\right)\mathrm{dy} \\ $$$$\mathrm{C}=\:\left(\mathrm{x}−\mathrm{a}\right)\mathrm{cos}\:\mathrm{a}+\mathrm{sin}\:\mathrm{a}\:\int\:\frac{\mathrm{d}\left(\mathrm{sin}\:\mathrm{y}\right)}{\mathrm{sin}\:\mathrm{y}}\: \\ $$$$\mathrm{C}=\left(\mathrm{x}−\mathrm{a}\right)\mathrm{cos}\:\mathrm{a}+\mathrm{sin}\:\mathrm{a}\:\mathrm{ln}\:\mid\mathrm{sin}\:\left(\mathrm{x}−\mathrm{a}\right)\mid\:+\mathrm{k}\: \\ $$

Commented by peter frank last updated on 02/Oct/22

thanks

$$\mathrm{thanks} \\ $$

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