微積分第5章習(xí)題選解

P198,xx（2）xx

2dx3

x2C3xexdxx

C 若fx)dx

Cfxx f(x)(x1)x

(x6.1,3),點處的切線的傾角為45o,求此曲線方程.解設(shè)曲線方程為yfx由題意,ykx所 f(x) kxdx1kx2C2因 f(1)ktan45o1,所1

f(x)1x2C,5

f(1)

C3,C

f(x)1x25 P218,（2）(32x)3dx1(32x)3d(32x)1(32x)4C d(2x (2x

2(2x

C23x22(223x2（6） dx1 d(23x2)1

C2323x23x23（7） dx dx43ln1x4C1x4 41 （8）xex2dx1ex2C2 tan10xsec2xdx tan10xdtanx1tan11xC cos23xdx1cos6xdx1x1sin6xCxx xxxsinxdxx

x

C x2sinxdxsinxd(x)cosxCtan3xsecxdxtan2xd(secx)(sec2x1)1sec3xsecxC3

4arctanx1x2

dx

1x2

dx 1x22(arctanx)21ln(1x2)C2tantan d(tanx)tantan

Ccos2xtan （18）xlnx1cos

lnlnxCd(xsin（20）xsinxdx

xsin

lnxsinxCeexxdxeexde

C8e4x2lnxdxe4x2xdx1e4x2C8 d(lnlnxlnxlnlnxlnxlnlnxlnln

lnlnlnxC11x

102

dx

102

darccosx

1022ln10

Cx xdx2 xx

x xx(1 1 x)2C（28） dx1sec2xdxtanxC.1cos

2cos2 2或解： 1cosxdx(csc2xcotxcscx)dx1cos sin2

1cos

sincotxcscxC 或

sin

C

或1cos

C3（30）cos3xdx(1sin2x)dsinxsinx1sin3xC3x （32）9x2dx2(19x2（34）dx1lnx2C

x ln(9

)C4x x （36）sin2xcos3xdx2(sin5xsinx)dx10cos5x2cosxC

d(sin2 （38）1sin4xdx1sin4x

x)C（40）dxcsc4xdx(1cot2x)d(cotx)cotx1cot3xCsin4 （41）1

t，x1t22

dxtdtt 1原式 dtt 1

tln1tC

ln1

C1 3x（42）3x

13xt，x1t3，dx3t3x2原式32

t21 1 dt dt3(t1 13

1

1 t23t3lnt1C2

(x1)32

x13ln

x11C23x（44） dx23xt2x1(2t2dx2td23x 原式2(2t2)tdt2(t22)dt2t34t

(23x)2

23xC222或解： dx123x222

22

(23x)2x1x1x

23xC

dxxsint原式sin2tcostdt1cos2tdt1t1sin2t 1x21arcsinx1x2

C2xx2xx2

xsect，原式sectsecttansecttan

或解：倒數(shù)代換x t原式

t(1t1t11t

dtsgn(t)dtsgn(t)arcsintsgn(x)arcsin1Cx

(x2

xtant 原式 sec2tdt costdtsint x1x1x

C.11x2

dxxtanttan3tsec21

dt1

tsectdtxx2

t sec3tsectC x

(x21)2x

C2x2x2

1

2dx2

dxxuI2uuu

u1uu3

du

1x2x2

)du

1(u1)2 uu3 (x21)23x

C（50）x1)10dxx1t原式(t1)2dt(

1t

t t

t7t

C

7(x

4(x

9(x

C22xx2令

x12

dx 9(x9(x1)2422原式3cost3costdx9cos2tdt91cos2t 9t

9sin2t

2x sint ，cost

2xx2 2x2x12xx2

9arcsin2x1C4或套 ：

a2a2xx2a2x

a99(x1)242

arcsinxC22xx222xx2

dx

dx2x

9arcsin2x1C55xx2

dx

2121(x1)2422x15xx2212x15xx2

1arcsin3x 4a2x2 arcsina2x2axx1(x142 xx1(x1421x x1x1原式1costdt(1

)dtttant 2111xt

1

Carcsinx

C

見P225xsinxdxxdcosxxcosxcosxdxxcosxsinxCxexdxxdexxexexdxxexex(x1)exCx2lnxdx1lnxdx31x3lnx1x31dx1x3lnx1x3C ln

lnxd1lnx dxlnx

(11(1

1

1

x(1

1

1ln1

1

Cxln2xdxxln2xx2lnx1dxxln2x2lnxxxln2x2xlnx2xC xcos2dx2xdsin22xsin22sin2dx2xsin24cos2C xsinxcosxdx2xsin2xdx4x1xcos2x1cos2xdx1xcos2x1sin2xC

xxln(x1)dx

ln(x1)dx2

ln(x1)2

x11x2ln(x1)1(x11)dx1x2ln(x1)1x21x1ln(x1)C

x1

1

x dxarctanxdx

x(1x2)dx2 dx1 dx21(11 )dx2

1

Cx(1x2

x2(1x2

x

1x

1x

arctanx1

C2 1x22（12）(x21)sin2xdx1(x221(x21)cos2x12xcos2x 1(x21)cos2x1x 1(x21)cos2x1xsin2x1sin2x 1(x21)cos2x1xsin2x1cos2xC xx

xdx

t,xt2，dx2tdtxx原式2tsintxx

2tdcost2tcost2cost

2tcost2sint

x

C3e3xdx3

t,xt3，dx3t2dt原式3t2etdt3t2det3t2et6tetdt3t2et6t3t2et6tet6etdt3et(t22t2)

3e3x

x2

x2)Cxcoslnxdxxcoslnxxsinlnx1dxxcoslnxsinlnxxxcoslnxxsinlnx xcoslnx1xxcoslnxxsinlnxcoslnxdxx移項,得coslnxdx2(coslnxsinlnxC

dx，令arctanxtxtantdxsec3(1x2)3

tdt原式tantetsec2tdtetsintdt1et(sintcostCsec3 x2x2由xtant，得cost ，sinttanx2x2原式12

xx2

Cfx的一個原函數(shù)為(1sinxlnxxfx)dx解xfx)dxxdfx)xfxfxf(x)(1sinx)lnxC1sin而fx

cosxlnxxxf(x)dxxf(x)(1sinx)lnx1sinxxcosxlnx(1sinx)lnxCP234x

x327 1.

x

dx

x

dx(x23x9

x

1x33x29x27lnx3C 2x dxd(x23x10)lnx23x10C.x23x x23xx5x4 x5x3x4x2x3xx2x3.

x3

dx

x3 (x2x1x2x8)dx

(x2x11 )dxx3

x x31令x3

Ax

x

x

有1Ax21Bxx1Cxx1x0A1x1B12x1,得C121x3

1x

2(x

2(x所以原式x2x1834 x x1x31x2x8lnx3lnx14lnx1C 3x31

解 x3 x

x2x得1Ax2x1(BxCx1x1A13x2B13；AC1C23.3所 3x3

x

x,x2x所以原式x

12x132 xx2lnx11ln(x2x1)3 (x1/2)23/lnx11ln(x2x1)2

3arctan2x1C33x(x2解

ABxCx(x2 x2有1Ax21x(BxC x0,得A1；取xi,

1BCi,B1,C0所以原式(1 )dxlnx1ln(x21)C x2

x(x2x2

x x2(x2

(1x2

x2

)dx2

12

xx2

C(x1)2(x1)x2 解設(shè) (x1)2(x1) x1 x1 (x1)2 x21A(x1)2B(x1)(x1)C(x1)x1A1x1,得C1x0,得1ABCB1x2(x1)2(x

,1 , 2x x1 (x 所以原式1dx11dx dx1lnx211C. x x

(x

x1(x21)(x2x1解 (x21)(x2x

AxBx2

Cx x2x有1AxBx2x1(CxDx21xi,得1ABiA1B0x0D1x1,得112(CDC1 x所以原式(x21x2x1ln(x21)1ln(x2x1)1 12

x2x1x2

313

2x33

C

(x1/2)23/(x21)(x21解 1(x21)(x2

A x

CxDx2有1Ax1x21Bxx21(CxD)xx1x0A1x1B12xi,得1(CiD)(1iCDi(CDCDCD

CD 121 1

1x1(x21)(x2

2(x

2x2所以原式[1 x1 2(x 2(x2lnx1lnx11ln(x21)1arctanxC

x

1

)dx2

x22

C(x21)(x2

6x2

x2

x

3xx(x

2x

x

(x

(x3x2dx2lnx2 Cx(xx

x

x1t

2(x 令2

t，有sinx

1x

1t

1t

12

dx

1

dt3cosx令tan2t，則cosx1t2，dx1t2dttan2222 原式 2dt dt 1arctantC1arctan 2C2222 42t 2tse

x

dtan

tan或解： 1 1 21arctan 2C22cos22

1cos22

se

x2

tan2x2

3sin2

sec23sec2xtan2x

dtan 4tan2x123 arctan2tanx1231sinxcosx

1t 令2

t，有sinx

1

1t

1

dt原式 2dt2t

lnt1

lntanx1C2

sec2

1 21sinxcos

tanx 2 2 dtanxlntanx1C.tanx1 ,2sinx

令tan2t，則sinx1t2，dx1t2dt原式

2dt

2t21t

1t

2(1t2)

t2t

2tanx3333 2arctan2t1C3333(t1/2)23/

2arctan C.x2

(x1)2

(x1)10dx

(x

dx(x1)8(x 7(x

9(x

Cx(1ln 1ln1ln2(1lnx)C23.

eexe

dx

e2e2x

dx

e2x

1ln(e2x1)C24.

2x3

2x

12x2x

2x31(2x3)23xx

1(2x1)2C335.

dx2arcsinx(1x1arcsinx(1x1

xd(arcsinx 3x5 25x9xdx

(5)35

dx

d()535()2x35

3

)2x3( ln3

C

55x35x3

C52ln53ex

()1ex1

2ln3

eex1dx

ex

dxx2ex1x21exx2ln(1ex)C 1sinxdx(sec2xsecxtanx)dxtanxsecxC.1sin cos2 tanxsec2xdx3sin2xcos2 3tan2x dtanx1 dtan2x1ln(tan2x1)C.3tan2x 3tan2x 110、x21dx x x4

x2x

(x1x

2 1212

Cx1 x1 xx1 x11、x22x5dxx1)24dxx1)241ln(x22x5)1arctan(x1)C 12

dx1 d(x21)1arctanx21C.x42x2 (x21)2 xx 13、 dx x3xx x

xx32(x3)xx33

C14、 dx令2x1t2

tdttln1t1 2x2x 2x

2x1)C

11x15、 令xsint dtcsc2tdt1x11x2x

C

sin2tx216、 令xsectsecttantdtcos2x2x31(1cos2t)dt2

1t2

1sin2tC

arccos1x

x22x2

C（1sin2t4

1sintcost2

12

cos2t

2sec2

）x2x21ln1ln1ln17、 ln dxlnx1ln1ln1ln

)dlnx1ln11ln (1lnx)23

C18

(1 ex/2e

ex/2(1ex/2)ex/2

ex/

1ex/

2

ex/2

dx

2

21)C19、xln(1x)dxdxln(1x)d1lnx1ln(1x) x x(1lnx1ln(1x)(11 1lnx1ln(1x)lnxln(1x)C(11)ln(1x)Cx20ln(1exdxln(1ex)de

ln(1ex)e e

1eln(1 )Cln(1ex)exdxln(1ln(1 )Ce 1ex e1x221(arcsinx)2dxx(arcsinx)22xarcsin1x21x21x2x(arcsinx)2arcsinxd(1x2)x(arcsinx)21x21x21x2x(arcsinx)1x21xx(arcsinx)1x

1x21x21x1x21x21x222lnx

1x2)dxxln(x

1x2) dxxln(x

1x2)1x1x

C

12x2112x21