第三章 不定积分
=?ln3x?2?ln2x?1?c
例3.21．?解：原式=
1dx
(x?1)2(x?1)111111?x?1?x1111?x??dx= =dx??ln?c 22??2x?121?x2(x?1)(x?1)2x?141?x例3.22．?解：原式=?1e2x?11dx
ex1?e?2xdx=?e?x1?e?2xdx=??de?x1?(e?x)2=?arcsine?x?c
例3.23．?sec3xtan3xdx
11解：原式＝?sec2xtan2xdsecx??(sec4x?sec2x)dsecx?sec5x?sec3x?C
53xarctan3x2dx 例3.24．?41?x1tan3x221132242dx?tanxd(arctanx)?arctan(x)?C 解：原式＝?21?x42?82．直接交换法 a）题型?f(ax?b)dx
(t2?b)方法：令t?ax?b，x?，
a2f(ax?b)dx?tf(t)dt ?a?1dx 例3.25．?x?1解：令t?x,x?t2，
1dt2tdt=2?dt?2?=2t?2lnt?1?c=2x?2ln(1?x)?c t?1t?11dx 例3.26．?x?2x?1?3原式=?解：令t?x?1,x?t2?1
原式=? 83 / 26
2tt?1?1112dt?2dt?d(t?1)??(t?1)2?3?(t?1)2?3?(t?1)2?3dt t2?2t?4第三章 不定积分
=ln(t2?2t?4)?例3.27．?311t?11x?1?1arctan()?C?ln(x?2x?1?3)?arctan()?C 3333dx
x?xx?t616t5t3)dt dt=6?(t2?t?1?解：原式6??23dt=6?x?tt?t1?t1?t=2t3?3t2?6t?ln(1?t)?c =2x?33x?66x?ln(1?6x)?c
例3.28．?1e?1xdx
解：原式
t?1?exx?ln(t2?1)?12t?t?t2?1dt
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 f(a2?x2)dx 变换x?atant
x2?a2)dx 变换x?asect
?f(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．?1x41?x2dx
84 / 26
第三章 不定积分
2sec2tcos3t1?sintdt?dt?解：令x?tant，原式???sin4t?sin4tdsint tan4t?sect1 =?csc3t?csct?C
3例3.31．?x3x?42dx
解：令x?2sect，
8sec3t42tantsectdt?8sectdt 原式???2tant8=8?(1?tan2t)dtant?8tant?tan3t?C（还原略）
3 例3.32．?1?1?x?23dx
2解：令x?tant，
原式??例3.33．?1x2sectdt?costdt?sint?c??c 3?2sect1?x1(x?2)x?2x?22dx
解：令x?1?tant， 1sint?costdcostdsintdt??2dt??原式＝?＝ 222??sint?costsint?cost1?2cost1?2sint ＝?122ln|1?2cost11?2sint。 |?ln||?C（还原略）
1?2cost221?2sint3．分部积分法 公式：?udv?uv??vdu 四种基本题型 a）题型1
??P?x?emxdx
例3.34．?(2x?1)edx 解：原式=
1112x2x2x(2x?1)de?(2x?1)e?ed2x 2?22?2x 85 / 26
第三章 不定积分
1 =(2x?1)e2x?e2x?C
2例3.35．?e2x?1dx
解：原式t?2x?1?ettdt??tdet?tet?et?C =2x?1e2x?1?e2x?1?C
例3.36．?xex22x2?x432dx
4?x2?1x2x42解：?xed????ed()2?2?242u??x221x21x2uu?2uedu?2e?2?ude?2e
444xxx1x21uu22=2ue?2e?e?C?xe?2e2?e2?C
222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 ??22233 =?xcos(2x?1)?sin(2x?1)?C
24例3.38．?xcos2xdx
1?cos2xx21dx???xdsin2x 解: 原式=?x?244x211??xsinx??sin2xdx 444x211 =?xsin2x?cos(2x)?C
448xdx 例3.39．?cos2x|C解： 原式＝?xdtanx?xtanx??tanxdx?xtanx?ln|cosx?
例3.40．?sin(x?1)dx
解：原式t?x?sin(t?1)2tdt??2?tdcos(t?1)??2tcos(t?1)?2?cos(t?1)dt
??2tcos(t?1)?2sin(t?1)?C??2xcos(x?1)?2sin(x?1)?C
86 / 26
第三章 不定积分
题型3
?x?xeecos?xdx或?sin?xdx ?例3.41．?e2xcos3xdx
112x32x2xcos3xde?ecos3x?esin3xdx ??22213139 =e2xcos3x??sin3xde2x?e2xcos3x?e2xsin3x??e2xcos3xdx?C
24244139 =e2xcos3x?e2xsin3x?C?I
24423 解得：I?e2xcos3x?e2xsin3x?C
1313解：设I??e2xcos3xdx?题型4
?P(mx)l?n(o)r(?arcta?n( dx例3.42．?xln(x?1)dx
1121x22dx 解：原式=?ln(x?1)dx=xln(x?1)??222x?1111)dx =x2ln(x?1)??(x?1?22x?11111 =x2ln(x?1)?x2?x?lnx?1?C
2422例3.43．?xln(x?1)dx
23ln(t?1)dt ?3221232t3)dt dt=t3ln(t?1)??(t2?t?1? =tln(t?1)??33t?133t?1232t312 =tln(t?1)?(?t?t?ln(t?1))?C
333223231222x?ln(x?1)?C =xln(x?1)?x2?x?39333解：原式?t?x?tln(t?1)2tdt?例3.44．??2x?1?ln2xdx
1解：原式??ln2xd(x2?x)?(x2?x)ln2x?2?(x2?x)lnxdx
xx22222 ?(x?x)lnx?2?(x?1)lnxdx?(x?x)lnx?2?lnxd(?x)
2x2x2122 ?(x?x)lnx?2(?x)lnx?2?(?x)dx
22x122x?c x?(x?2x)lxn?2x2? ?(x2?x)ln2 87 / 26