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

Prove that:  if (x∈]−(π/2),(π/2)[  y =∫^( x) _( 0) (dt/(cos(t))) ) ⇒  (y∈IR   x =∫^( y) _( 0) (dt/(cosh(t))) )

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

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