西南交大高数自测题答案第四章

第四章

1 x

t，得x ln t2 1 ，故

x

t2 1 ln t2 1 tln t2 1 t2 1 ln t2 1 2t

tt 12

tln t2 1 tdln t2 1 2 ln t2 1 dt 2

2t2 t2 1 1 22 2 tln t 1 2dt 2 tln t 1 2 2 t 1 t 1

1 2 tln t2 1 2 dt 2 2 t 1

2 2 tlnt 1 2t 2arctant C1

2tln t2 1 4t 4arctant

C

2C

2求 arctanxdx 22x(1 x)

arctanxarctanxarctanx 1 arctanxd dx dx arctanxdarctanx x2(1 x2) x2 1 x2 x

arctanx11 arctanx arctan2x xx2arctanx11 arctan2x 2x2x1 x arctanx1x 1 arctan2x dx 2 x2x1 x

arctanx11 arctan2x lnx ln 1 x2 C x22

lnsinx3求 2 sinx

lnsinx2 sin2x lnsinxcscxdx lnsinxd cotx cotxlnsinx cotxd lnsinx

cos2x cotxlnsinx cotxdx cotxlnsinx 2sinx2

1 sin2x cotxlnsinx cotxlnsinx csc2x 1 dx 2sinx

cotxlnsinx cotx x C

x 54求 2 x 6x 13

1 2x 6 8x 512x 68dx 22 x2 6x 13 x2 6x 13 2x 6x 13x 6x 13

1182x 6x 13 22 2x 6x 134 x 3

1x 3 ln x2 6x 13 4arctan C 22

lnx5设f lnx ，计算 f x dx x

t解：令t lnx，故f t t， e

x故 f x dx x xe xdx xd e x e

xe x e xdx xe x e x C

arctanex

6求 e2x

arctanex1 2x 1 2x1 2xx x arctaned e earctane edarctanex e2x 22 2 1 2x111 2x1e x

xx earctane earctane 22 ex1 e2x221 e2x

1 2x1 1ex

x earctane x dx 22 e1 e2x

11 1x e 2xarctanex e xdx 2x 22 1 e

111 e 2xarctanex e x arctanex C 222

7设f sin2x

xf x dx

，求sinx sint 0 t ，则t

cost

2

故sint x dx f sin2t dsin2t 2 f sin2t sin2tdt cost 2 tsintdt 2 td cost 2tcost 2 costdt 2tcost

2sint C C

arcsinex8求 ex

arcsinex

xxx x earcsined arcsined e ex

e xarcsinex

e xd arcsinex

earcsine xx x

xdx e xarcsinex

令t x ln 1

t2 1

2

t1 112 dln1 t 2 t 2 1 tt1 t 2

11 11 11 t C

C dt ln22 1 t1 t 21 t

1arcsinex

xxC earcsine 故 2e

x

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