Prove-that-if-x-pi-2-pi-2-y-x-0-dt-cos-t-y-IR-x-y-0-dt-cosh-t-

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

Prove that :

if (x \in] - \frac{π}{2}, \frac{π}{2} [y = {\int_{0}}^{x} \frac{dt}{\cos (t)}) \Rightarrow (y \in IR x = {\int_{0}}^{y} \frac{dt}{\cosh (t)})

Answered by MrGaster last updated on 03/Feb/25

Let y = \int_{0}^{x} \frac{d t}{\cos (t)} \Rightarrow y = \ln (\tan (\frac{x}{2} + \frac{π}{4}))

Then x = 2 \int_{0}^{y} \frac{d t}{e^{t} + e^{- t}} \Rightarrow x = 2 \int_{0}^{y} \frac{d t}{2 \cosh (t)}

Thuzs x = \int_{0}^{y} \frac{d t}{\cosh}

Conclusion : y \in R x = \int_{0}^{y} \frac{d t}{\cosh (t)}

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