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Question Number 144142 by bobhans last updated on 22/Jun/21

$$\int \frac{\sqrt{\cos \mathrm{x}+\sqrt{\cos \mathrm{x}+\sqrt{\cos \mathrm{x}+\sqrt{\cos \mathrm{x}+\sqrt{...}}}}}}{\sin \mathrm{x}}\mathrm{dx}$$

Answered by liberty last updated on 22/Jun/21

$$\sqrt{\cos \mathrm{x}+\sqrt{\cos \mathrm{x}+\sqrt{\cos \mathrm{x}+\sqrt{...}}}}=\ell$$$$\sqrt{\cos \mathrm{x}+\ell }=\ell \Rightarrow {\ell }^2-\ell -\cos \mathrm{x}=0$$$$\Rightarrow \ell =\frac{1+\sqrt{1+4\cos \mathrm{x}}}{2}$$$$\mathrm{I}=\int \frac{1+\sqrt{1+4\cos \mathrm{x}}}{2\sin \mathrm{x}}\mathrm{dx}$$

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