Bonjour,

On effectue le changement de variable u = tan(t) du = (1+tan²(t))dt = (1+u²)dt = (u²+1)dt
I = ∫ tan⁵(t) dt = ∫ u⁵/(u²+1) du

Or, u⁵ = u⁵+u³-u³-u+u = u³(u²+1)-u(u²+1)+u

I = ∫ u⁵/(u²+1) du = ∫ [u³-u+[u/(u²+1)]] du = u⁴/4 - u²/2 + (1/2)ln(u²+1) + K = tan⁴(t)/4 - tan²(t)/2 + (1/2)ln(tan²(t)+1) + K

Merci Marcel =)