高数。要过程。

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换元

这两个题有一定联系的。如图。

见图

偶倍奇零
原式=∫(-1/2,1/2)|ln(1-x)|dx

∫(0,1/2)|ln(1-x)|dx=-∫(0,1/2)ln(1-x)dx+∫(-1/2,0)ln(1-x)dx
=-xln(1-x)|(0,1/2)+∫(0,1/2)x/(x-1)dx
=-1/2×ln(1/2) +∫(0,1/2)[1+1/(x-1)]dx
=1/2ln2+[x+ln|x-1|]|(0,1/2)
=1/2ln2+[1/2+ln1/2]-0
=1/2ln2+1/2-ln2
=1/2-1/2ln2
∫(-1/2,0)ln(1-x)dx
=xln(1-x)|(-1/2,0)-∫(-1/2,0)x/(x-1)dx
=1/2ln3/2 -∫(-1/2,0)[1+1/(x-1)]dx
=1/2ln3/2 -(x+ln(1-x))|(-1/2,0)
=1/2ln3/2-[0-(-1/2+ln3/2)]
=1/2ln3/2 -1/2+ln3/2
=-1/2+3/2ln3/2
相加，得
原式=-1/2ln2+3/2ln3/2
=-1/2ln2+3/2ln3/2
=-1/2ln2+3/2ln3+3/2ln1/2

=3/2ln3+ln2