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

$$\int \frac{\sin \mathrm{x}}{\sin (\mathrm{x}-\mathrm{a})}\mathrm{dx}$$

Answered by cortano1 last updated on 02/Oct/22

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

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

$$\mathrm{thanks}$$

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