Question

∫dx／（(x²＋1)³)½

Answer 1

令x=tant 则(1+x^2)^3/2=(sect)^3 dx=(sect)^2dt
∫dx/(1+x^2)^3/2 =∫costdt=sint+c=x/(1+x^2)^1/2+c

Answer 2

∫dx／（(x^2＋1)^3)^(1/2)
=∫dx/(x^2＋1)^(3/2)
=x/√(x^2+1) +C

let
x=tanu
dx=(secu)^2 du

∫dx/(x^2＋1)^(3/2)
=∫(secu)^2 du/(secu)^3
=∫ cosu du
= sinu + C
=x/√(x^2+1) +C