第三章 不定积分
xx??sin?cos??22?x?（39）原式=?e?dx
?x?2cos2???2??0.5?ex(tan ?exd?tan2xxx?1)2dx?0.5?ex(sec2?2tan)dx 222???x?xxxxxxxx?etandx?etan?etandx?etandx?C ????2?2222?extan（40）原式=x?C 21??1dx??dxex??lnxex?ln1?xex?C xx???xex?1?xex????xe1?xe?x?3x?3222，u?，xu?u?x?3 x?1x?1?x?1?ex（41）令u?u2?3?8u,dx?du 所以x?222u?1?u?1?1??8uu2?11?du??8du??8du 原式=??u??222??22uu?1???u?1??u?1? ?4ln 本章测试答案 1?u?C?4ln1?ux?1?x?3?C x?1?x?323xcosx?Cxe?C 2. 3. 3xcos2x2?C??cotx?C 4. ?2sinxx2x2x21?x)?(?x)lnx??(?x)?dx 5．原式=?lnxd(222xx2x2?(?x)lnx??x?C
24lnu22udu?4?lnudu?4ulnu?4u?C?4xlnx?4x?C 6．原式u?x?u1. - 103 -
7．原式
x?2sint1?4sin2t?2cost?2costdt ?1?cot(t)?C 4?14?x2???C
4x2sinxcosxsinxdx?2?cos2x?cosxdx??2lncosx?C
x9. tanx??C
28．原式=
10.?2sinx?C
5211.(xlnx)2?C
5x2?412.ln(x?4?x)??C
x212x?ln|2sinx?cosx|?C 55sinxxcosx?sinx)??, 14.证明：f(x)?(2xx13.?3332 xf?(x)dx?xdf(x)?xf(x)?3xf(x)dx
???2＝x(xcosx?sinx)?3xdsinx?3sinxdx?xcosx?4xsinx?6cosx?C
??15.xf??(2x?1)dx??x2x＝2＝16.原
11??xf(2x?1)d(2x?1)?xdf?(2x?1) ??221x1f?(2x?1)??f?(2x?1)dx?f?(2x?1)??f?(2x?1)d(2x?1)
2241f?(2x?1)?f(2x?1)?C
4xln(x?x2?1)??xx2?1dx?xln(x?x2?1)?x2?1?C
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