tanx?1?tanx?tan2xdx 解 令tanx?u
tanxu111dx?du?du??1?tanx?tan2x?1?u?u21?u2?1?u2?1?u?u2du
⒁
?arctanu?22u?122tanx?1arctan?C?x?arctan?C 3333x2(1?x)2?2(1?x)?1⒂ ?dx??dx 100100(1?x)(1?x)?111???C
99(1?x)9949(1?x)9897(1?x)97⒃
arcsinx1arcsinx1dx??arcsinxd???dx ?x2??2xxx1?xarcsinx1?1?x2???ln||?C
xx⒄
?xln1?x1dx??x[ln(1?x)?ln(1?x)]dx??[ln(1?x)?ln(1?x)]dx2 1?x2121211x2?11?x?x[ln(1?x)?ln(1?x)]??x(?)dx?ln?x?C 221?x1?x21?x⒅
?1sinxcosx7dx??1tanxcosx4dx??1?tan2xtanxdtanx
1?2tanx(1?tan2x)?C
521?x2ex?2xx1?x?2xx⒆ ?e()dx?edx?dx?edx 222222???1?x(1?x)1?x(1?x)xex1exexexexx??dx??ed??dx???dx??C 2222221?x1?x1?x1?x1?x1?x⒇ In??vnudx，u?a1?b1x，v?a2?b2x
172
解 In??vnudx?2n2nvdu?[vu?n?ub2vn?1dx] ?b1b1b2uvn?1(b1v?a1b2?a2b1)vn?12n2n?[vu?n?dx]?[vu?n?dx] b1b1uu?2n[vu?nb1In?n(a1b2?a2b1)In?1] b12[vnu?n(a1b2?a2b1)In?1]
(2n?1)b1所以In? 173