13(2x?1)?31?x?212?3??2??2x?1x?1x?x?1x?113(x?)2?()222 1(2x?1)1312???x?1(x?1)2?3213(x?)2?()224221(2x?1)3131??3dx??dx??2dx??dx13x?1x?1213(x?)2?(x?)2?()224221x?111312)?lnx?1??d((x?)2?)?3?d(12(x?1)2?3243x?242)2?12(3212x?1?lnx?1?ln(x2?x?1)?3arctan()?C.23
★★★(4)?x?1dx (x?1)3思路：将被积函数裂项后分项积分。 解：令
x?1ABC???(x?1)3x?1(x?1)2(x?1)3，等式右边通分后比较两边分子x的同次项的系
数得：
A?0,B?2A?1,?A?B?C?1，解此方程组得：A?0,B?1,C?2。
x?112??(x?1)3(x?1)2(x?1)3
x?11211x??dx?dx?dx????C???C32322??x?1(x?1)(x?1)(x?1)(x?1)(x?1)★★★(5)?3x?2dx 3x(x?1)思路：将被积函数裂项后分项积分。 解：
3x?232??x(x?1)3(x?1)3x(x?1)3，令
2ABCD????x(x?1)3xx?1(x?1)2(x?1)3
等式右边通分后比较两边分子x的同次项的系数得：
A?B?0??A?2?3A?2B?C?0?B??2??解此方程组得：??3A?B?C?D?0??C??2??A?2??D??2?22222????x(x?1)3xx?1(x?1)2(x?1)3。
26
3x?2322221222?????????x(x?1)3(x?1)3xx?1(x?1)2(x?1)3(x?1)3xx?1(x?1)23x?21222??dx?dx?dx?dx?dx332????x(x?1)(x?1)(x?1)x?1x112????2lnx?1?2lnx?C22(x?1)x?1??2lnx4x?3??C.x?12(x?1)2xdx(x?2)(x?3)2
★★★(6)?
思路：将被积函数裂项后分项积分。 解：
?xx?2?2x?22???(x?2)(x?3)2(x?2)(x?3)2(x?2)(x?3)2(x?2)(x?3)2
，等式右边通分后比较两
12?(x?3)2(x?2)(x?3)2；令
2ABC???(x?2)(x?3)2x?2x?3(x?3)2边分子x的同次项的系数得：
A?B?0??A?22222??6A?5B?C?0B??2????解此方程组得：??22x?2x?3(x?2)(x?3)(x?3)?9A?6B?2C?2?C??2??
x1222322??(??)???(x?2)(x?3)2(x?3)2x?2x?3(x?3)2(x?3)2x?2x?3xdx322???dx?dx?dx 22???(x?2)(x?3)(x?3)x?2x?3?33?x?3????2lnx?2?2lnx?3?C?ln???C.?x?3x?2x?3??2★★★(7)?3xdx x3?1思路：将被积函数裂项后分项积分。
3x3(x?1)?333??? 3323x?1x?1x?x?1x?1?C令33?A?Bx，等式右边通分后比较两边分子x的同次项的系数得： x?1x?1x2?x?1解：
?A?B?0?A?1??A?B?C?0 解此方程组得：??B??1?A?C?3?C??2???
31?x?21x?2???? x3?1x?1x2?x?1x?1x2?x?1131313(2x?1)?(2x?1)(2x?1)2?2222而2x?2?22 ???2222x?x?1x?x?1x?x?1x?x?1x?x?1x?x?1 27
33x11(2x?1)??3dx??22dx??dx??2dxx?1x?x?1x?12x?x?11x?11 22)?lnx?1?1?3?d(d(x?x?1)?x2?x?1123x?2)2?12(32?3arctan2x?11?lnx?1?ln(x2?x?1)?C
23x?12x?1?3arctan?ln?C 23x?x?11?x?x2★★★(8)?22dx
(x?1)思路：将被积函数裂项后分项积分。 解：
1?x?x21x2????(x2?1)2x2?1(x2?1)2(x2?1)2
1?x?x21xdx??2dx??dx?dx?2?x2?1?(x2?1)2?(x2?1)2(x?1)2111dx2???2dx??2d(x?1)?2?(x2?1)22(x?1)2x?1
又由分部积分法可知：2?dxx1??dx (x2?1)2x2?1?x2?11?x?x2x1112x?1??2dx???C?(2)?C 222(x?1)x?12x?12x?1★★★(9)?xdx
(x?1)(x?2)(x?3)思路：将被积函数裂项后分项积分。 解：令
xx?3?313???
(x?1)(x?2)(x?3)(x?1)(x?2)(x?3)(x?1)(x?2)(x?1)(x?2)(x?3)3ABC???，
(x?1)(x?2)(x?3)x?1x?2x?3等式右边通分后比较两边分子x的同次项的系数得：
3?A?33??A?B?C?02?3?2?3?2 5A?4B?3C?0B??3??解之得：??(x?1)(x?2)(x?3)x?1x?2x?3?6A?3B?2C?3?3??C?2?而
111??
(x?1)(x?2)x?1x?2 28
3x112?????2(x?1)(x?2)(x?3)2x?1x?2x?3xdx11dx3dx?????dx?2??? (x?1)(x?2)(x?3)2x?1x?22x?313??lnx?1?2lnx?2?lnx?3?C.22x2?1★★★(10)?dx 2(x?1)(x?1)思路：将被积函数裂项后分项积分。 解：令
x2?1x2?1?212 ???222(x?1)(x?1)(x?1)(x?1)x?1(x?1)(x?1)2ABC???(x?1)2(x?1)x?1x?1(x?1)2，等式右边通分后比较两边分子x的同次项的系数
得：
11A?B?0,2A?C?0,A?B?C?2；解之得：A?,B??,C??1。
221122?2?1??(x?1)2(x?1)x?1x?1(x?1)211x?12?2?1??(x?1)2(x?1)x?1x?1(x?1)22
x2?11dx1dx1??dx???dx 22???(x?1)(x?1)2x?12x?1(x?1) ?1lnx?1?1lnx?1?22111?C ?lnx2?1??C. x?12x?1★★★(11)?1x(x2?1)dx
思路：将被积函数裂项后分项积分。 解：令
1x(x2?1)?ABx?C?2，等式右边通分后比较两边分子x的同次项的系数得： xx?1?A?B?0?A?111x???? ?C?0解之得：?B??1?22x(x?1)xx?1?A?1?C?0????1x11dx?dx?dx?lnx?d(x2?1)222???x2x?1x(x?1)x?1x11?lnx?ln(x2?1)?C?ln?C.22x?1
★★★(12)?dx 22(x?x)(x?1) 29
思路：将被积函数裂项后分项积分。 解：令
11 ?(x2?x)(x2?1)x(x?1)(x2?1)1ABCx?D???(x2?x)(x2?1)xx?1x2?1，等式右边通分后比较两边分子x的同次项的系数
得：
A?B?C?0,A?C?D?0,A?B?D?0,A?1，解之得：
111A?1,B??,C??,D??.
22211111x?1?2?????(x?x)(x2?1)x2x?12x2?111111x11 ???????(x2?x)(x2?1)x2x?12x2?12x2?1dx1111x1dx??2?dx?dx?dx?2?x?12?x2?12?x2?1(x?x)(x2?1)?x?1111?lnx?lnx?1??2d(x2?1)?arctanx24x?12
111?lnx?lnx?1?ln(x2?1)?arctanx?C.242★★★★★(13)?dx x4?1思路：将被积函数裂项后分项积分。 解：令
x4?1?(x2?1?2x)(x2?1?2x)
1Ax?BCx?D??x4?1x2?1?2xx2?1?2x，等式右边通分后比较两边分子x的同次项的系数
得：
?2?A??4?A?C?0?1??B???2A?B?2C?D?02解之得：????C?2?A?2B?C?2D?0??4B?D?1???D?1??2?
112x?212x?22(2x?2)?22(2x?2)?2??????4x2?1?2x4x2?1?2x88x4?1221221(x?)?(x?)?22222(2x?2)(2x?2)111?[?]?[?]84221221221221(x?)?(x?)?(x?)?(x?)?22222222dx2(2x?2)(2x?2)111??4?[?]dx?[?]dx??84x?1221221221221(x?)?(x?)?(x?)?(x?)?22222222
30