Question

对根号(x平方-a平方)/x如何积分?

Answer 1

令x=asecm

则分子=atanm

dx=a*secmtanmdm

secm=x/a

cosm=a/x

所以m=arccos(a/x)

(tanm)^2=x^2/a^2-1=(x^2-a^2)/a^2

所以tanm=√(x^2-a^2)/a

所以原式=∫(atanm/asecm)a*secmtanmdm

=∫a(tanm)^2dm

=a∫[(secm)^2-1]dm

=a∫(secm)^2dm-a∫dm

=atanm-am+C

=√(x^2-a^2)-a*arccos(a/x)+C