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Question Number 95562 by peter frank last updated on 26/May/20

$$\mathrm{show}\mathrm{that}$$$$\int \frac{\sin (\mathrm{x}-\theta )}{\sin \mathrm{x}}\mathrm{dx}=\mathrm{xcos}\mathrm{x}-\sin \theta \log \sin \mathrm{x}$$

Answered by Ar Brandon last updated on 26/May/20

$$\mathbf{sin}(\mathbf{x}-\mathit{\theta })=\mathbf{sin}\left(\mathbf{x}\right)\mathbf{cos}\left(\mathit{\theta }\right)-\mathbf{cos}\left(\mathbf{x}\right)\mathbf{sin}\left(\mathit{\theta }\right)$$$$\Rightarrow \mathrm{I}=\int \frac{\mathbf{sin}\left(\mathbf{x}\right)\mathbf{cos}\left(\mathit{\theta }\right)-\mathbf{cos}\left(\mathbf{x}\right)\mathbf{sin}\left(\mathit{\theta }\right)}{\mathbf{sin}\left(\mathbf{x}\right)}\mathbf{dx}=\int \mathbf{cos}\left(\mathit{\theta }\right)\mathbf{dx}-\int \frac{\mathbf{cos}\left(\mathbf{x}\right)}{\mathbf{sin}\left(\mathbf{x}\right)}\mathbf{sin}\left(\mathit{\theta }\right)\mathbf{d}\mathit{\theta }$$$$=\mathbf{xcos}\left(\mathit{\theta }\right)-\mathbf{sin}\left(\mathit{\theta }\right)\mathbf{log}\left(\mathbf{sin}\left(\mathbf{x}\right)\right)+\mathbf{C}$$

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