第三章 不定积分
=-4?(x?1)?arcsin例3.19．
2x?1?c 2?1?e?dxx2
x2x2解：原式=
1?e?ex21?e1?e1例3.20．?dx （2x?1)(3x?2)323(2x?1)?2(3x?2)dx??dx 解：原式=??dx=??3x?22x?1(2x?1)(3x?2)=?ln3x?2?ln2x?1?c 例3.21．dx=x?2?dex2x2=x?2ln(1?e)?c
x21?(x?1)2(x?1)dx 111111?x?1?x1111?x??dxdx= =??ln?c 2x?12?1?x22?(x?1)2(x?1)2x?141?x解：原式=例3.22．?1e2x?11dx 解：原式=?ex1?e?2xdx=?e?x1?e?2xdx=??de?x1?(e)?x2=?arcsine?x?c 33例3.23．secxtanxdx ?解：原式＝secxtanxdsecx?(secx?secx)dsecx??22?42151secx?sec3x?C 53xarctan3x2dx 例3.24．?1?x41tan3x221132242dx?tanxd(arctanx)?arctan(x)?C 解：原式＝?421?x2?82．直接交换法
a）题型
?f(ax?b)dx
(t2?b)方法：令t?ax?b，x?，
a- 83 -
?例3.25．
f(ax?b)dx?2tf(t)dt ?a?1dx x?1解：令t?原式=
x,x?t2，
1dt2tdt2dt?2=?t?1??t?1=2t?2lnt?1?c=2x?2ln(1?x)?c 1例3.26．?dx
x?2x?1?3解：令t?x?1,x?t2?1
2tt?1?1112dt?2dt?d(t?1)??t2?2t?4?(t?1)2?3?(t?1)2?3?(t?1)2?3dt
原式=
=ln(t2?2t?4)?1t?11x?1?1arctan()?C?ln(x?2x?1?3)?arctan()?C 3333dx
例3.27．
x?xx?t616t5t3)dt dt=6?(t2?t?1?解：原式6??23dt=6?x?tt?t1?t1?t=2t?3t?6t?ln(1?t)?c =2x?33x?66x?ln(1?6x)?c
例3.28．
32?13?1e?1xdx
解：原式
12t??2dt x?ln(t2?1)tt?1t?1?ex?1?ex?111?t)?c dt=ln=2?2?c=ln(xt?11?t1?e?1b) 题型
?f(ax2?b)dx
?
f(a2?x2)dx 变换x?asint
a2?x2)dx 变换x?atant
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?f(第三章 不定积分
?f(x2?a2)dx 变换x?asect
例3.29．
?9?x2dx x解：令x?3sint，
3cost11?sin2t3costdt=3?dt?3?sintdt dt=3?原式??3sintsintsint9?x21??31?cost?39?x23 =ln?3cost?C=ln??C 221?cost239?x1?3例3.30．?x141?x2dx sec2tcos3t1?sin2tdt??4dt??dsint 解：令x?tant，原式??44tant?sectsintsint13 =?csct?csct?C 3例3.31．?x3x?42dx 解：令x?2sect， 8sec3t42tantsectdt?8sectdt 原式???2tant832=8?(1?tant)dtant?8tant?tant?C（还原略） 3 例3.32．?1?1?x?23dx 2解：令x?tant， 原式?1x2sectdt?costdt?sint?c??c ?sec3t?21?x例3.33．
?(x?2)1x?2x?22dx
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解：令x?1?tant，
原式＝
1sint?costdcostdsintdt?dt??＝?sint?cost?sin2t?cos2t?1?2cos2t?1?2sin2t
＝?122ln|1?2cost11?2sint。 |?ln||?C（还原略）
1?2cost221?2sint3．分部积分法
公式：udv?uv?四种基本题型 a）题型1
??vdu
?xPxedx ???m2x例3.34．(2x?1)edx
?1112x2x2x(2x?1)de?(2x?1)e?ed2x ??22212x2x =(2x?1)e?e?C
2解：原式=例3.35．e解：原式t??2x?1dx
2x?1?ettdt??tdet?tet?et?C
2x?1=2x?1e例3.36．xex22?e2x?1?C
?x2?x432dx
4?x2?1x2x42解：?xed????ed()2?2?24u??2x221x21x2uu?2uedu?2e?2?ude?2e
2444xxx1x21uu22=2ue?2e?e?C?xe?2e2?e2?C
22题型2
?P(x)cos?xdx或?P(x)sin?xdx
mm例3.37．3xsin(2x?1)dx 解：原式=
??333xdcos(2x?1)??xcos(2x?1)?cos(2x?1)dx 2?22?- 86 -
第三章 不定积分
=?33xcos(2x?1)?sin(2x?1)?C 242例3.38．xcosxdx
?1?cos2xx21dx???xdsin2x 解: 原式=?x?244x211??xsinx??sin2xdx 444x211?xsin2x?cos(2x)?C =448xdx 例3.39．?cos2x解： 原式＝xdtanx?xtanx?例3.40．sin(x?1)dx 解：原式t???tanxdx?xtanx?ln|cosx?|C ?x?sin(t?1)2tdt??2?tdcos(t?1)??2tcos(t?1)??2cos(t?1)dt ??2tcos(t?1)?2sin(t?1)?C??2xcos(x?1)?2sin(x?1)?C 题型3 ?x?xeecos?xdx或?sin?xdx ?2x例3.41．ecos3xdx ?112x32x2xcos3xde?ecos3x?esin3xdx ???22212x312x32x92x2x =ecos3x??sin3xde?ecos3x?esin3x??ecos3xdx?C 2424412x32x9 =ecos3x?esin3x?C?I 24422x3ecos3x?e2xsin3x?C 解得：I?1313解：设I?ecos3xdx?2x题型4 ?P(mx)l?n(o)r(?arctan?( )dx例3.42．xln(x?1)dx
?1121x22dx 解：原式=?ln(x?1)dx=xln(x?1)??222x?1- 87 -