1/(x^2*sqrt(x^2+1)）的积分

1/(x^2*sqrt(x^2+1)）的积分
1/(x^2*sqrt(x^2+1)）的积分

1/(x^2*sqrt(x^2+1)）的积分
令x=tany,则siny=x/√(x²+1),dx=sec²ydy
∫dx/[x²√(x²+1)]=∫sec²ydy/(tan²y*secy)
=∫cosydy/sin²y
=∫d(siny)/sin²y
=C-1/siny (C是积分常数)
=C-1/[x/√(x²+1)]
=C-√(x²+1)/x.

令 x = tant, dx = sec²t dt， sqrt(x^2+1) = sect
I = ∫ sect / tan²t dt = ∫ cost /sin²t dt = -1/ sint + C
= - sqrt(x^2+1) / x + C