Mon calcul est juste ?

[Anonyme]

Bonjour, voici le calcul :

Calculer ;) x²/(;)1+x²) dx avec x = sht

x = -ch(t)
x²-1 = ch²(t) - 1 = sh²(t)
x² = ch²(t)

dx = -sh(t) dt

S [x²/V(x²-1)] dx = - S [(ch²(t)/sh(t)).sh(t)] dt = - S ch²(t) dt

ch(t) = (1/2).(e^t + e^-t)
ch²(t) = (1/4).(e^2t + 2 + e^-2t)

S [x²/V(x²-1)] dx = - (1/4). S e^2t dt - (1/2). S dt - (1/4). S e^-2t dt

S [x²/V(x²-1)] dx = - (1/8 ).e^2t - (1/2).t + (1/8 ).e^-2t

S [x²/V(x²-1)] dx = -(1/2).t - (1/8 ).[e^2t - e^-2t]

S [x²/V(x²-1)] dx = -(1/2).t - (1/4).sh(2t) + K

S [x²/V(x²-1)] dx = -(1/2).argch(-x)) - (1/4).sh(2.argch(-x))

S (x²/Vx²+1) dx = 1/8 e^2t – ½ t – 1/8 e^-2t

S (x²/Vx²+1) dx = - ½ t + 1/8 (e^2t – e^-2t)

S (x²/Vx²+1) dx = - ½ t + ¼ ch(2t)

S (x²/Vx²+1) dx = - ½ arg sh(x) + ¼ ch (2arg sh(x))


Merci !

[Anonyme]

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