?ln?e?1?xe?xxdx???ln?e?1?d?exx?x???ex?xln?e?1??x?e1x?1dx?x??e??eln?e?1??x?1?ee?xdx??e?x?x?xln?e?1???1?e1d?1?e?x?
?xln?e?1??ln?1?e??C.edx1x?2xx（3）
?xe2x2?x?2?xdx?ex??x?2?2?4?x?2??42?x?2?ex2?e?4?xx?2xdx?4?ex?x?2?2dx?e?4?ex2xde?4?4exxex2?x?2??C.dx
?e?4ex?2?4??x?2?dx?4??x?2?dx?e?x?2（4）
?lnsinxsinx2dx???lnsinxdcotx??cotxlnsinx??cot2xdx
??cotxlnsinx???cscx4332x?1?dx??cotxlnsinx?cotx?x?C.（5）
???x435dx?313x?1?x?1d?x1333???331x?1?14143x?133d?x?1?3134??x3?1?4d?x?1??37??x?1??d?x?1?
21?x3?1?4?49?x3?1?4?C.（6）
第 6 页 共 8 页
?arctanxx(1?x)22dx??arctanxx2dx??arctanx1?x2dx2?1?1???arctanxd????arctanx??x?2??arctanxxarctanxxarctanxx??12121x?1?x2?dx?12?arctanx?22
2????ln1x2?1?x?2d?x???arctanx?221???x221?x?12?arctanx??C.（7）
?e?x2cosx?sinxsinx?x2dx???ex2?x2cosxsinxdx??x2?e?x2sinxdx?x2?x2?2?e?2e?dsinx??esinxdx?2esinx??esinxdx??esinxdx
x2sinx?C.（8）
??x?1x?1?xex?dx???x?1?exexxx?1?xe?dx??1xex?1?xexexxx?d?xex?
?1xed?xexx???1?xe1d?1?xecx??ln1?xe?C.（9）
x?1x?146?dx???x6?x2???x42?1?1?dx?1x2x?1?x2dx??x?121x2dx
2?C.2?x33??x??31?x1?xd?x????ln21x?322?1??x???x???2xx??1(10)
第 7 页 共 8 页
?arcsin2x1?x2dx?xarcsinx2x1?xx2??1?xdxxx?xarcsin1?x?2??1??1?xx?2??d?xarcsin2x1?x2x?2???x
?1?1???2dx?xarcsin1?x?2x?2arctanx?C.?1,十、设f?lnx????x,\0?x?1,1?x???及f?0??0,f?0??1,求f?x?的表达式。
?t?C1,，积分得f?t???t0?t????e?C2,''?1,解：令t?lnx，则f?t???t?e,\???t?0,???t?0,0?t???
???t?0,?t?1,由f?0??1,可知f?t???t
e,0?t????''?t?t?C1,???t?0,?再积分得f?t???2由f?0??0,
?et?C,0?t????2?t?t,?可知f?t???2?et?1,?22???t?0,0?t???
第 8 页 共 8 页