Integral de e ^ ((x + 2) ^ 1 / 2)∫e ^ ((x + 2) ^ 1 / 2) dx?

∫e ^ (√(x + 2)) dx

Sea t = √(x + 2)

t² = x + 2

2t dt / dx = 1

2t dt = dx

∫e ^ t * 2t dt

Recordemos que :

∫u dv = uv - ∫v du

Sea u = 2t

du / dt = 2

du = 2dt

Sea dv = e ^ t dt

∫dv = ∫ e ^ t dt

v = e ^ t

∫e ^ t * 2t dt = 2te ^ t - ∫e ^ t2dt = 2te ^ t - 2∫e ^ tdt = 2te ^ t - 2e ^ t + c = 2e ^ t(t - 1) + c = 2 e ^ (√(x + 2)) (√(x + 2) - 1) + c.