对根号(x平方-a平方)/x如何积分?是(x平方-a平方)开平方再除以x,然后求定积分!别弄混淆啦!

x = |a|sect, 0 dx = |a|sect*tantdt.
S [x^2 - a^2]^(1/2)dx/x
= S |a||tant|*|a|sect*tantdt/[|a|sect]
= |a|S |tant|*tantdt
0 |a| > 0.
S [x^2 - a^2]^(1/2)dx/x
= |a|S |tant|*tantdt = |a| S [tant]^2 dt = |a| S {[sect]^2 - 1}dt
= |a|{tant - t} + C
= |a|{tan[arccos(|a|/x)] - arccos(|a|/x)} + C
PI/2 S [x^2 - a^2]^(1/2)dx/x
= |a|S |tant|*tantdt = -|a| S [tant]^2 dt = -|a| S {[sect]^2 - 1}dt
= -|a|{tant - t} + C
= -|a|{tan[arccos(|a|/x)] - arccos(|a|/x)} + C

令x=a/sint 去做

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