∫In(x+√(1+x^2))dx解题过程

Question

∫In(x+√(1+x^2))dx解题过程

Answer 1

u = In(x+√(1+x^2))
e^u = x + √(1+x^2)
e^(-u) = √(1+x^2) - x
相减得
x = (e^u - e^(-u) ) / 2 = sh(u)
dx = ch(u)du
代入得
∫In(x+√(1+x^2))dx
=∫ u ch(u)du = ∫ u dsh(u) = ush(u) - ∫ sh(u)du = ush(u)-ch(u) + C
=x In(x+√(1+x^2)) - √(1+x^2) + C