f(n)1n!1(?1)n (x)????2(1?x)n?12(1?x)n?1 f(2k?1)(0)?0, k = 0, 1, 2, … f2k(0)?n!, k = 0, 1, 2, … 六. 设y?xlnx, 求f(n)(1).
(n?1)!n?2(n?2)!?n(?1) nn?1xx解. 使用莱布尼兹高阶导数公式 f(n)(x)?x?(lnx)(n)?n(lnx)(n?1)?x(?1)n?1n?2 =(?1)n?1??(n?1)n?2 (n?2)!???(?1)(n?2)!n?1n?1?n?1xxx??所以 f(n)(1)?(?1)n?2(n?2)! 七. 已知
?y0edt??costdt?siny2,求y'.
0y222t2x2解. 两边对x求导, ey'?2xcosx?2yy'cosy,y'?2xcosx2ey2?2ycosy2
第三章 一元函数积分学(不定积分)
一. 求下列不定积分: 1.
11?xln?1?x21?xdx
211?x11?x1?x1?1?x?lndx?解. ?lndln??ln??c ?1?x21?x21?x1?x4?1?x?11?x1?x1?x1?1?x?2. ?arctandx?arctandarctan?arctan???c 2?1?x1?x1?x2?1?x?1?x3.
2cosx?sinx?11?sinx?(1?cosx)2?1?cosxdx
2cosx?sinx?11?sinx1?sinx1?sinx1?1?sinx?解. ??dx???c ?1?cosxd1?cosx?2?(1?cosx)21?cosx1?cosx??4.
?dx 8x(x?1) 11
1解. 方法一: 令x?,
t?dxt7dt1?dt????ln(1?t8)?c 88??1?1x(x?1)t?18??8?1?t?t?1?1?ln?1?8??c 8?x??1t2 = ?
111(1?sinx?cosx)?(sinx?cosx)?1?sinx22dx 5．?dx??21?sinx?cosx1?sinx?cosx11cosx?sinx11dx??dx ??dx??221?sinx?cosx21?sinx?cosx11d(1?sinx?cosx)11 ?x????dx
xxx221?sinx?cosx22sincos?2cos22221111x ?x?ln|1?sinx?cosx|??dtan
x2222tan?12111x ?x?ln|1?sinx?cosx|?ln|tan?1|?c
2222二. 求下列不定积分: 1.
?(x?1)dx2x?2x?22
解.
?(x?1)2dt2dxd(x?1)cost 令x?1?tant ???tan2tsectx2?2x?2(x?1)2(x?1)2?1costdt1x2?2x?2 =????c???c
sin2tsintx?12.
?xdx41?x2
解. 令x = tan t,
?x4dt2dxcos3tdsintdsint11cost???dt??????c 44423???2tantsectsintsintsint3sintsint1?x 12
321?1?x =??3?x?2?1?x???c
?x?3.
?(2xdx2?1)1?x2
解. 令x?tant
sec2tcostdsint??dt?dt? ? 2222??22(2tant?1)sect2sint?cost1?sint(2x?1)1?xdx =arctansint?c?arctanx1?x2?c
4.
??x2dxa?xx2dx22 (a > 0)
解. 令x?asint
a2sin2t?acostdt1?cos2t11???a2?dt?a2t?a2sin2t?c
acost224a2?x2?a2?x2??c
?a2?xx =?arcsin?22?aa5.
??(1?x2)3dx
解. 令x?sint
(1?cos2t)21?2cos2t?cos22t(1?x)dx??costdt??dt??dt
44234111311t?sin2t??(1?cos4t)dt?t?sin2t?sin4t?c 4488432311 =arcsinx?sin2t(1?cos2t)?c
844 =
314?1?2sin2t)?c =arcsinx?2sintcost(844 =arcsinx?381x1?x2(5?2x2)?c 86.
?x2?1dx x41 t解. 令x? 13
?x2?1dx??4x1?t2t21t4?1?22??2?dt???t1?tdt 令t?sinu??sinucosudu ?t?(x2?1)313 =cosu?c??c
33x37.
?xx?12x?12dx
dx?secttantdt
解. 令 x?sect,
?xx?12x2?1dx??sect?1secttantdt??(1?cost)dt?t?sint?c 2secttantx2?1?c x1 ?arccos?x三. 求下列不定积分:
e3x?exdx 1. ?4xe?e2x?1e3x?exex?e?xd(ex?e?x)x?x解. ?4xdx?dx??arctan(e?e)?c 2x2x?2xx?x2??e?e?1e?1?e(e?e)?12.
dx?2x(1?4x)
dt tln2x解. 令t?2, dx?
dxdt1?11?1arctant???dt????c ???2x(1?4x)?t2(1?t2)ln2ln2??t21?t2?tln2ln21(2?x?arctan2x)?c ln2 =?
四. 求下列不定积分:
x51. ?dx
(x?2)100x51x555?994?99解. ?dx??xd(x?2)???x(x?2)dx 10099??(x?2)9999(x?2)99 14
x55x45?4 =???x3(x?2)?98dx 9998?99(x?2)99?98(x?2)99?98x55x45?4x35?4?3x2 =? ???9998979699(x?2)99?98(x?2)99?98?97(x?2)99?98?97?96(x?2) ?5?4?3?2x5?4?3?2???c 959499?98?97?96?95(x?2)99?98?97?96?95(x?2)
2.
?x?dx1?x4解.
1dt2dxtdt1dt2t令x?1/t???????
444222x1?x1t?11?t1?(t)tt4?1sec2u111?x4令t?tanu??du??ln|tanu?secu|?c??ln?c
2secu22x22
五. 求下列不定积分: 1.
2xcosxdx ?2xcosxdx??1121x(1?cos2x)dx?x??xdsin2x 2?441211 ?x?xsin2x??sin2xdx
4441211 ?x?xsin2x?cos2x?c
448解. 2.
3sec?xdx
解.
?sec3xdx??secxdtanx?secxtanx??tanxsecxtanxdx
=secxtanx?(secx?1)secxdx?secxtanx?ln|secx?tanx|?secxdx
3sec?xdx??2?311secxtanx?ln|secx?tanx|?c 22(lnx)3dx 3. ?2x(lnx)3113(lnx)233dx???(lnx)d??(lnx)??dx 解. ?x2xxx2 15