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Question Number 162112 by amin96 last updated on 26/Dec/21

$$\int \frac{\mathit{c}\mathit{o}\mathit{s}\left(\mathit{x}\right)}{{(1-\mathit{c}\mathit{o}\mathit{s}\left(\mathit{x}\right))}^2}\mathit{d}\mathit{x}=?$$

Answered by Ar Brandon last updated on 26/Dec/21

$$I=\int \frac{\cos x}{{(1-\cos x)}^2}dx=\int \frac{{\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2}}{{4\sin }^4\frac{x}{2}}dx$$$$=\frac{1}{4}\int {\cot }^2\frac{x}{2}{\mathrm{cosec}}^2\frac{x}{2}dx-\frac{1}{4}\int {\mathrm{cosec}}^2\frac{x}{2}dx$$$$=\frac{1}{2}\cot \frac{x}{2}-\frac{1}{6}{\cot }^3\frac{x}{2}+C$$

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