x5?x4?88432??x?x?1???xx?1x?1x3?xx5?x4?88432??dx?(x?x?1???)dx ?xx?1x?1x3?x11?x3?x2?x?8lnx?4lnx?1?3lnx?1?C32★★★(3)
3?x3?1dx
思路：将被积函数裂项后分项积分。 解：?x3?1?(x?1)(x2?x?1)，令
3ABx?C??等式右边通分后比较两边分子32x?1x?1x?x?1x的同次项的系数得：
?A+B=0?A?1???B+C-A=0解此方程组得：?B??1?A+C=3?C?2??
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)31312??3dx??dx??dx??dx123x?1x?1213(x?)?(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?1?(x?1)3dx
思路：将被积函数裂项后分项积分。
31
解：令
得：
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
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???C?(x?1)2?(x?1)3x?1(x?1)2(x?1)3(x?1)2?★★★(5)
3x?2?x(x?1)3dx
思路：将被积函数裂项后分项积分。 解：?3x?2322ABCD，令??????x(x?1)3(x?1)3x(x?1)3x(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
。
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??2ln★★★(6)
x4x?3??C.x?12(x?1)2xdx?(x?2)(x?3)2
思路：将被积函数裂项后分项积分。 解：?xx?2?2x?22??? 2222(x?2)(x?3)(x?2)(x?3)(x?2)(x?3)(x?2)(x?3) 32
?122ABC；令，等式右边通????2222x?2x?3(x?3)(x?3)(x?2)(x?3)(x?2)(x?3)分后比较两边分子x的同次项的系数得：
A?B?0??A?22222????? ?6A?5B?C?0解此方程组得：?B??2?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?3?x?2?x?3★★★(7)
23x?x3?1dx
思路：将被积函数裂项后分项积分。
3x3(x?1)?333???
x3?1x3?1x2?x?1x3?13ABx?C??2令3，等式右边通分后比较两边分子x的同次项的系数得：
x?1x?1x?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)x?22?2222而2 ?22???2222x?x?1x?x?1x?x?1x?x?1x?x?1x?x?133x11(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
23 33
?3arctanx?12x?1?ln?C
23x?x?11?x?x2★★★(8)?(x2?1)2dx
思路：将被积函数裂项后分项积分。
1?x?x21x2解：?2????(x?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???(x2?1)2x2?1?x2?1dx
1?x?x2x1112x?1??2dx???C?()?C
(x?1)2x2?12x2?12x2?1★★★(9)
xdx?(x?1)(x?2)(x?3)
思路：将被积函数裂项后分项积分。
?解：
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)令
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?33?5A?4B?3C?0?2??2 解之得：??B??3?(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 34
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)?(x?1)2(x?1)dx
思路：将被积函数裂项后分项积分。
x2?1x2?1?212解：? ???(x?1)2(x?1)(x?1)2(x?1)x?1(x?1)2(x?1)令
2ABC，等式右边通分后比较两边分子x的同次项的系数得： ???(x?1)2(x?1)x?1x?1(x?1)211A?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) ?11111lnx?1?lnx?1??C ?lnx2?1??C. 22x?12x?1★★★(11)
?x(x112?1)dx
思路：将被积函数裂项后分项积分。 解：令
x(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?0B??1???解之得： ??22x(x?1)xx?1?A?1?C?0?? 35