求下列不定积分（根号x+1-1）/（根号x＋1＋1）dx

Question

求下列不定积分（根号x+1-1）/（根号x＋1＋1）dx

Answer 1

令t=√(x+1) 那么x=t^2-1
dx=2tdt
(√(x+1)-1)/(√(x+1)+1)·dx
=((t-1)/(t+1))2t·dt
=(1-2/(1+t))2t·dt
=(2t-4t/(1+t))·dt
=(2t-(4t+4)/(1+t)+4/(1+t))·dt
=(2t-4+4/(1+t))·dt
=t^2 - 4t + 4ln|1+t| + c
=x+1-4√(x+1)+4ln(1+√(x+1))+c
=x-4√(x+1)+4ln(1+√(x+1))+c