工科数学分析教程上册最新版习题解答5.33

5.3典型计算题三

计算下列定积分：

1、881(2)(2)88

x

x x e dx x e d x +=+⎰⎰⎰+=x de x 8)2(81 8811(2)88x x x e e dx =+-⎰8811(2)864x x x e e C =+-+ 2、arcsin 2xdx ⎰ dx x x

x x ⎰--=24122arcsin ⎰--=22412arcsin x dx x x

C x x x x x d x x +--=---=⎰22

241212arcsin 41)41(412arcsin 3、dx x x xdx x ⎰⎰+⋅=22cos 1cos 2⎰+=dx x x x )2cos (2

1 ⎰+=x xd x 2sin 4

1412 2111sin 2sin 2444

x x x xdx =+-⎰ 2111sin 2cos 2448

x x x x C =+++ 4、dx x arctg ⎰-13 令t x =-13，则312

t x +=，dt t dx 32= ⎰⎰=⋅=23

132arctgtdt dt t arctgt dt t t arctgt t ⎰+-=2

2

213131 ⎰+--=dt t

arctgt t )111(313122 C arctgx t arctgt t ++-=3

131312 C x arctg x x arctg x +-+----=133

1133113)13(31 5、⎰⎰⎰-----=---=-x x x de x x d e x dx e x 222)1(2

1)2()1(21)1( C e e x x x +---=--224

1)1(21 6、⎰⎰-=-x d x xdx x 4cos )43(4

14sin )34(

⎰--=xdx x x 4cos 4

34cos )43(41 C x x x +--=4sin 16

34cos )43(41 7、dx x )9ln(2

+⎰ ⎰+-+=dx x x x x 92)9ln(22

2 2229ln(9)2(1)ln(9)269

3x x x dx x x x arctg C x =+--=+-+++⎰ 8、⎰

dx x arctg ，令t x =，则2t x =，tdt dx 2= 22arctgt tdt arctgtdt =⋅=⎰⎰

⎰+-=dt t t arctgt t 22

2

1C arctgt t arctgt t ++-=2 C x arctg x x xarctg ++-=

9、⎰⎰-=-x x de x dx e x 44)52(4

1)52( ⎰--=

dx e e x x x 4424

1)52(41 C e e x x x +--=448

1)52(41 10、⎰⎰-=dx x x xdx x 24cos 12sin 21(cos 4)2

x x x dx =-⎰ 221111cos 4sin 44248

x x xdx x xd x =-=-⎰⎰ ⎰+-=xdx x x x 4sin 8

14sin 81412 C x x x x +--=4cos 3214sin 81412 11、2

ln(21)x dx +⎰ dx x x x x ⎰+-+=124)12ln(22

2 2

2

22ln(21)221x x x dx x =+-+⎰ ⎰+--+=dx x x x )121

1(2)12ln(22

⎰++-+=1

222)12ln(22x dx x x x C x arctg x x x ++-+=222)12ln(2

12、dx x arctg ⎰-14，令t x =-14，则4

12

t x +=，dt t dx 2= ⎰⎰=⋅=24

12arctgtdt dt t arctgt t d t t arctgt t ⎰+-=2

2

214141 222111111(1441444

t arctgt dt t arctgt t arctgt C t =--=-+++⎰

11(41)44

x C =-+ 13、⎰⎰+=⋅+x x d x dx x 2)25(2

ln 12)25( 25(52)2ln 2ln 2

x x x dx =+-⎰ 225(52)2ln 2ln 2

x x x C =

+-+ 14、⎰⎰=43ln 4

1ln xdx xdx x ⎰⋅-=dx x x

x x 44141ln 41 C x x x +-=4416

1ln 41 15、⎰⎰-=-)35sin(5

1)35cos(x xd dx x x ] ⎰---=dx x x x )35sin(5

1)35sin(51 C x x x +-+-=)35cos(251)35sin(51 16、⎰-dx x )25arccos( 先分部积分再令t x sin 25=-

sin 2t

+dt t cos )25arccos(-=x x dt t t t 5cos cos 5sin 25⋅⋅++⎰

⎰

++-=dt t x x )sin 2(51)25arccos( C t t x x +-+-=cos 5152)25arccos( 25-x

C x x x x +----+-=2)25(15

1)25arcsin(52)25arccos( 17、⎰⎰=323

1arctgxdx arctgxdx x dx x

x arctgx x ⎰+-=23313131 222216131dx x x arctgx x ⎰+-=⎰+--=223)111(6131dx x

arctgx x 322111ln(1)366

x arctgx x x C =-+++ 18、⎰⎰=2arcsin 2

1arcsin xdx xdx x dx x

x x x ⎰--=222121arcsin 21

2

11arcsin 22x x dx =

+⎰

2111arcsin arcsin 222

x x x =+- sin ,cos x t dx tdt ==令则 ⎰++-=dt t x x x )2cos 1(4

1arcsin 21arcsin 212 C t t x x x +++-=2sin 8

141arcsin 21arcsin 212

21111arcsin arcsin arcsin 2244

x x x x C =-++ 19、⎰⎰+=+x d x xdx x 5sin )32(5

15cos )32( ⎰-+=xdx x x 5sin 5

25sin )32(51 12(23)sin 5cos5525

x x x C =+++ 20、⎰⎰-=⋅-x x d x dx x 3)1(3

ln 13)1( ⎰-⋅-=dx x x x 33ln 13)1(3

ln 1 C x x x +--=33

ln 13)1(3ln 12 21、⎰⎰--+-=+x x de x dx e x 22)4(2

1)4( ⎰--++-=dx e e x x x 222

1)4(21