令x = tanz,dx = sec²z dz,secz = √(1 + x²)
∫ x³√(1 + x²) dx
= ∫ tan³z * |secz| * sec²z dz
= ∫ tan³zsec³z dz
= ∫ tan²zsec²z d(secz)
= ∫ (sec²z - 1)sec²z d(secz)
= ∫ (sec⁴z - sec²z) d(secz)
= (1/5)sec⁵z - (1/3)sec³z + C
= (1/5)(1 + x²)^(5/2) - (1/3)(1 + x²)^(3/2) + C