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解;换元法:
令t=(1-x)^1/2
3/4<=x<=1
t属于[0,1/2]
t^2=1-x
x=1-t^2
dx=(-2t)dt
原式=积分0 1/2 (-2t)dt/(t-1)
=-2积分0 1/2 tdt/(t-1)
t/(t-1)=(t-1)+1/(t-1)=1+1/(t-1)
原式=-2(积分1dt+积分1/(t-1)dt)
=-2x(1/2+ln/1/2-1/-ln/0-1/)
=-2x(1/2-ln2-0)
=-1+2ln2
=2ln2-1
答:元积分值是2ln2-1.
令t=(1-x)^1/2
3/4<=x<=1
t属于[0,1/2]
t^2=1-x
x=1-t^2
dx=(-2t)dt
原式=积分0 1/2 (-2t)dt/(t-1)
=-2积分0 1/2 tdt/(t-1)
t/(t-1)=(t-1)+1/(t-1)=1+1/(t-1)
原式=-2(积分1dt+积分1/(t-1)dt)
=-2x(1/2+ln/1/2-1/-ln/0-1/)
=-2x(1/2-ln2-0)
=-1+2ln2
=2ln2-1
答:元积分值是2ln2-1.
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