(lnx)33(lnx)26lnx(lnx)33(lnx)26lnx6???2dx??????2dx ??xxxxxxx(lnx)33(lnx)26lnx6????c ??xxxx4.
?cos(lnx)dx
解.
?cos(lnx)dx?xcos(lnx)??sin(lnx)dx?x[cos(lnx)?sin(lnx)]??cos(lnx)dx ?x[cos(lnx)?sin(lnx)]?c 2 ?cos(lnx)dx?5.
xcos4xxcos32211xx1x1x?2x??sin?2d??xsin?2?cot?c ??xsin824228242sin3六. 求下列不定积分: 1.
?x2dx?18?sin3xxcos4x2dx??111?2x?2x?2xxdsin??xsin?sindx ??828282?xln(x?1?x2)dx 22(1?x)解.
?xln(x?1?x2)112 dx?ln(x?1?x)d(1?x2)22?1?x211111ln(x?1?x2)??dx 22?221?x21?x1?x =
ln(x?1?x2)111 令x?tant ???sec2tdt 22?2(1?x)21?tantsectln(x?1?x2)1cost =?dt 22?2(1?x)21?2sintln(x?1?x2)1d2sint = ?2?2(1?x2)1?2sint22ln(x?1?x2)11?2sint =?ln?c 22(1?x)421?2sintln(x?1?x2)11?x2?2x?ln?c =222(1?x)421?x?2x
16
2.
?xarctanx1?x2dx
1?x2dx 21?x解.
?xarctanx1?x22dx??arctanxd1?x?1?xarctanx??22 =1?xarctanx??11?x2dx?1?x2arctanx?ln(x?1?x2)?c
arctanexdx 3. ?e2xarctanex11?2x1?2xexx?2xxdx???arctanede??earctane??edx 解. ?2x2xe2221?e1?2x111?2x1e?xxxdx??earctane?dx ??earctane??2xx2x?221?e22e(1?e)1?2x11ex1?2xxx?x)dx??(earctane?e?arctanx)?c ??earctane??(x?2x22e1?e2?xln(1?x2)?3x?0七. 设f(x)??2 , 求?f(x)dx. ?xx?0?(x?2x?3)e解.
??(xln(1?x2)?3)dx??f(x)dx??
2?x(x?2x?3)edx???12?122xln(1?x)?[x?ln(1?x2)]?3x?cx?0? ??2 2x?02?x??(x?4x?1)e?c?1考虑连续性, 所以
c =－1+ c1, c1 = 1 + c
?12?122xln(1?x)?[x?ln(1?x2)]?3x?cx?0?f(x)dx??2 2x?02?x??(x?4x?1)e?1?c?x
八. 设f'(e)?asinx?bcosx, (a, b为不同时为零的常数), 求f(x). 解. 令t?e，x?lnt, f'(t)?asin(lnt)?bcos(lnt), 所以
x 17
f(x)?[asin(lnx)?bcos(lnx)]dx =
?x[(a?b)sin(lnx)?(b?a)cos(lnx)]?c 2九. 求下列不定积分: 1.
32x4?xdx ?解. 令x?2sint
323222x4?xdx?32sintcostdt??32(1?cost)costdcost ???3233214 =?cost?cos5t?c?(4?x2)2?(4?x2)2?c
35532.
53??x2?a2dx(a?0) x解. 令x?asect
x2?a2atantdx(a?0)??asecttant?a?tan2tdt?atant?at?c xasect22 =x?a?aarccosa?c x3.
?ex(1?ex)1?e2xdx
ex1?e2x解.
?ex(1?ex)1?e2xd??dx+?e2x1?e2xdx
=
?dex1?e2x1d(1?e2x)－?dx=arcsinex?1?e2x?c 21?e2x4.
?xxdx (a > 0)
2a?xxu4解. ?xdx 令u?x 2?du 令u?2asint 8a2?sin4tdt
2a?x2a?u2(1?cos2t)2dt?2a2?(1?2cos2t?cos22t)dt =8a?421?cos4ta422dt?3at?2asin2t?sin4t?c =2at?2asin2t?2a?24222 =3at?4asintcost?asintcost(1?2sint)?c
18
2222
=3at?3asintcost?2asintcost?c =3aarcsin22223xx2a?xx?3a2?2a22a2a2a2ax3a?x?x(2a?x)?c 2a2x2a?x?c
2a2a =3aarcsin2十. 求下列不定积分:
2?sinx?2?cosxdx 2?sinx1d(2?cosx)dx?2?dx??解. ?
2?cosx2?cosx2?cosx2dt2x2dt1?t 令tan?t 2??ln|2?cosx|?2?ln|2?cosx| 22?21?t3?t2?1?t21. =
4t41xarctan?ln|2?cosx|?c?arctan(tan)?ln|2?cosx|?c
23333sinxcosx?sinx?cosxdx
sinxcosx11?2sinxcosx?1dx??dx 解. ?sinx?cosx2sinx?cosx2.
1(sinx?cos)2?1111dx??(sinx?cosx)dx??dx =?2sinx?cosx22sinx?cosx)124 =(sinx?cosx)??24sin(x??)4 =
d(x??12x?(sinx?cosx)?ln|tan(?)|?c 24282十一. 求下列不定积分: 1.
x3??3x(2x?3)dx
223x?3xx?3xx?3x2(2x?3)dx??3d(x?3)??c 解. ?3ln322.
?(3x2?2x?5)(3x?1)dx
232332122解. ?(3x?2x?5)(3x?1)dx??(3x?2x?5)2d(3x?2x?5)
2
19
12 ?(3x?2x?5)2?c
53.
5?ln(x?1?x2)1?x2dx
12ln(x?x2?1)?c 2解.
?ln(x?1?x2)1?x2dx??ln(x?x2?1)dln(x?x2?1)?
4.
?(1?xxdx2?x?1)ln(1?x?1)xdx222解.
?(1?x?x2?1)ln(1?x2?1)??dln(1?x2?1)ln(1?x2?1)?ln|ln(1?x2?1)|?c
十二. 求下列不定积分: 1.
xarctanx?(1?x2)dx
xarctanx1arctanx122?1dx?d(1?x)??arctanxd(1?x) 22?(1?x2)2??2(1?x)21arctanx111arctanx11?darctanx???dx
21?x22?1?x221?x22?(1?x2)2解.
??1arctanx11arctanx11?cos2t2?costdt????dt
21?x22?21?x2221arctanx111aextanx11?t?sin2t?c???arctanx?sintcost?c ??21?x24821?x2441aextanx11x?arctanx??c ??2221?x441?x 令x?tant?2.
?arcsinxdx 1?xx?t,1?x则x?tan2t
解. 令arcsin
?arcsinxdx??tdtan2t?ttan2t??tan2tdt?ttan2t?tant?t?c 1?xxxx?x?arcsin?c?(1?x)arcsin?x?c 1?x1?x1?x ?xarcsin 20