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Question Number 210566 by Erico last updated on 12/Aug/24

$$\mathrm{Prove}\mathrm{that}:$$$$\mathrm{if}(\mathrm{x}\in ]-\frac{\pi }{2},\frac{\pi }{2}[\mathrm{y}={\underset{0}{\int }}^{\mathrm{x}}\frac{\mathrm{dt}}{\cos \left(\mathrm{t}\right)})\Rightarrow (\mathrm{y}\in \mathrm{IR}\mathrm{x}={\underset{0}{\int }}^{\mathrm{y}}\frac{\mathrm{dt}}{\cosh \left(\mathrm{t}\right)})$$

Answered by MrGaster last updated on 03/Feb/25

$$\mathrm{Let}y={\int }_0^x\frac{dt}{\cos \left(t\right)}\Rightarrow y=\ln \left(\tan (\frac{x}{2}+\frac{\pi }{4})\right)$$$$\mathrm{Then}x=2{\int }_0^y\frac{dt}{e^t+e^{-t}}\Rightarrow x=2{\int }_0^y\frac{dt}{2\cosh \left(t\right)}$$$$\mathrm{Thuzs}x={\int }_0^y\frac{dt}{\cosh }$$$$\mathrm{Conclusion}:y\in \mathbb{R}x={\int }_0^y\frac{dt}{\cosh \left(t\right)}$$

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