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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))) )
Provethat:if(x]π2,π2[y=0xdtcos(t))(yIRx=0ydtcosh(t))
Answered by MrGaster last updated on 03/Feb/25
Let y=∫_0 ^x (dt/(cos(t)))⇒y=ln(tan((x/2)+(π/4)))  Then x=2∫_0 ^y (dt/(e^t +e^(−t) ))⇒x=2∫_0 ^y (dt/(2 cosh(t)))  Thuzs x=∫_0 ^y (dt/(cosh))  Conclusion:y∈R x=∫_0 ^y (dt/(cosh(t)))
Lety=0xdtcos(t)y=ln(tan(x2+π4))Thenx=20ydtet+etx=20ydt2cosh(t)Thuzsx=0ydtcoshConclusion:yRx=0ydtcosh(t)

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