北大版高等数学第三章 积分的计算及应用答案 习题3.3

1、习题 3.3:x 1x 11. x 2 6 x 8dx ( x 2 )( x 4 )dx, x 1( x 2 )( x 4 ) 21Ax B,x 4 415A 2 4 , B 2, 4 22 x 1( x 2 )( x 4 )x 1 3 /x 23 5 / 2 ,x 45dx ln | x 2| | x 4 | C .x 2 6 x 8222. I 3 xx 1dx.x 2 x 63 xx 1 3 x 2 3 x 2 2 4 0 x 1 3 3,x 2 x 640 x 13x 2 x 6 4 0 x 1 33AB,x 2 x 6( x 3)( x 2)x 3x 2A 4 0 ( 3) 1 3

2、3 2 5 3 , B 4 0 2 1 3 3 5 3 . 3 23 x 252 5 32 355 3I x 3 2 2 x 2ln | x 3 | ln | x 2 | C .553. I 2 x 2 5dxx 4 5 x 2 62 x 2 52 u (u x 2 )x 4 5 x 2 6u 2 5 u 62 u 5AB,(u 2 )( u 3)u 2u 3A 2 2 3 1, B 23 2 1.22 x 2 5211,x 4 5 x22I 122 6x 2 2x 2x x 22x 2 1x 23x 23x 33 23C.34. I dx.( x 1) 2 ( x 2 ) 111( x 1)

3、 2 ( x 2)x 2 x 2x 1 1 11,( x 2 )2 x 2x 1x 1x 2I x 1x 2x 2C.5. I x 2x 21 x 4dx.x 21(1 x 2 ) (1 x 2)1 x4(1 x 2 )(1 x2 )2(1 x 2 )(1 x 2 ) 1 11,2 1 x21 x2I 1 41 1 dx1 arctanx C .26. I .x 3 1 1 1AB x C,x 3 1( x 1)( x 2 x1)x 1x 2 x 1A 1 1 ,12 1 1x 2 x 1 33 ( x 1)( B x C ) ( B 1 ) x3 ( B C 11 ) x C ,33C 13

4、 1, C 21, B 3 0, B 1 .3 11 x 21x 2x 3 13( x 1)3( x 2 x1)3( x 1)3( x 2 x 1)12 x 411(2 x 1) 3.3( x 1)6 ( x 2 x1)3( x 6( x 2 x 1)112 x 1113,33( x 1)6( x x 1)21 22 x 32 23I 3ln | x 1 | 1ln(x6 x 1) 13arctan32 x 1C.7. I dx1. 111 x 4 1 x (1 2 x 2 x 4 ) 2 x2( x 2 1) 2 2 x 2( x2 2 ( x2 2 x 1)( x 2 2 x 1)x2 2

5、 x 1A x BC x D,x2 2 x 11 ( A x B x2 2 x 12 x 1) (C x D )( x 2 2 x 1),1 ( A C ) x 3 A C 0 ( B 2 A D 2C )x ( A 2 B C 2 D ) x B D . B A2 A D 2 B C 2 C 0,2 D 0,B D 1.B22A 1221, B 2, C 1, D 1 .22222122x 2 x 122x 2 x 122 x 1122x 2 x 122 x 11 x 4x 2xx 2x2 2 x 1x 2x2 2 x 12222422 x 22x2 422 x 22x2 2 x 12 x

6、22x2 2 x 1(2 x x 2 (2 x x 2 2 ) 2 x 12(2 x x 2 2 ) 2 x 124242(2 x 2)(2 x 2)x2 2 x 1( 2x 2)x2 2 x 124212 424212( x 12) 2 2 1 41.21122( x ) 2 242I 142x2 2 x 12 x2 2 x x2 2 x 14arctan (2 x 1) arctan (2 x 1) C .8. I x 3 x 2 2dx.( x 2 2 ) 2x 3 x 2 2 x ( x 2 2 ) x 2 x 2( x 2 2 ) 2( x 2 2 ) 2( x 2 2 ) 2x12

7、x.( x 2 2)( x 2 2)( x 2 2 )211x122I ln ( x 2 2) arctan C .222x 2 2exdxdexdu9.e2 3e x 2e2 3 e x 2u 2 3u 2du 11 du u C e x C.(u 1)( u 2) u 1u 2 u 2e x 210. cos xdxsin 2 x sin x 6 d sin xsin 2 x sin x 6(u sin x )duu 2 u 6dudu 1 11du ln C ln C .(u 3)( u 2) u 2u 3u 2u u 2u 3sin x 2sin x 3x 3 dx1x 2 dx1ud

8、u11. x 4 x 2 22x 4 x 2 22u 2 u 2 1 2udu 14u 2 u 24(2 u 1) 1du u 2 u 2 1 d (u 2 u 2)1du 1du24u 2 u 2241 7u u 72 47 1 ln (u 2 u 2 ) 747127arctan272 u 1 C 1 ln ( x 4 x 2 2 ) 4127arctan272 x 2 1C.dx1 2. I .dx( x 2 )( x 2 2 x 2 ) 1AB x C( x 2 )( x 2 2 x 2)x 2x 2 2 x 2A 11 .(2)2 2(2) 210 11( x 2 )( x 2 2

9、x 2)1 0 ( x 2) B x C x 2 2 x 21 0 ( x 2 2 x 2 )1 0 ( x 2 )( x 2 2 x 2 )B x C x 2 2 x 2 ( x 2 2 x 8)1 0 ( x 2 )( x 2 2 x 2 )B x C x 2 2 x 2 ( x 2 )( x 4)1 0 ( x 2 )( x 2 2 x 2) B x C x 2 2 x 2 ( x 4)B x C, B 12,C .1 0 ( x 2 2 x 2 )x 2 2 x 2105I 11 0ln | x 2 | 11 0 x 4dxx 2 2 x 21 ln | x 2 | 1 10102 x

10、 8dxx 2 2 x 21 ln | x 2 | (2 x 2 ) 6dx10211x 2 2 x 23dxln | x 2 |1 01ln(x2 01 2 x 2 ) 13 ( x 1) 2 1ln | x 2| ln ( x 2 2 x 2) arctan ( x 1) C1020101 3. I dx2 sin x. x u , x 2arctanu , dx 22 du 1 u, sin x 2 tanx22u.x1 u 22 duI 1 u2 1du 11 tan 2 23du32uu 2 u 21 221 u 2u 22 u12333arctan C 333arctan2 tan

11、2 2x 132 C .31 4. I .dx sin x co s xdxx u , x 2arctanu , dx 2du 1 u 2sin x 2u1 u 2, co s x 1 u2.1 u2I 2 du1 u 2 2u1 u1 2 11 u 2 2 u 1 u 2du 1du u 11 u21 u2x ln | u 1 | C ln | tan 1 | C 215.co txd x co t 2x2 x 1)dx co t 2x csc xd x co txd x co t 2xd co t x 2 x 1)dx 13co tx co t x x C .1 6. secxd x (1

12、 tan2 x )d tan x tan x 1tan33 x C .1 7. I co s xdx1 5 3 co s x33 co s xd x 5 3 co s x 1 ( 3 co s x 5) 5 dx35 3 co s x x 3xdx5 3 co s x2 du1 u 2tan u , dx2 , co s x1 u22du,1 u 2xI 5 1 u 2x5 2 du335 3(1 u2 )31 u 25(1 u 2 ) 3(1 u 2 ) x 5 2 du x 5 dux 5 1 d 2 u338 u 2 ) 334 u 2 334 u 2 1x 35 arctan2 u C

13、 6x 3arctan 2 tanx C . 2 dx1 8. I dxco s 3 xd x co s 2 xd x.sin x co s x1 tan du(1 tan x )(1 tan 2 x )tan x u , x arctanu ,dx ,1 u 2duI 1 u2(1 u )(1 u 2 ) du,(1 u )(1 u 2 ) 2 1 1 11 u (1 u )(1 u 2 ) 22 (1 u 2 ) 1 u1 u 2 1 11 u 1 u,4 1 u1 u 12 (1 u 2 )21111 1uI ln | 1 tan x | arctanu ln (1 u 2 ) arct

14、anu C441x184 (1 u 2 112 22 (1 u 2 )ln | 1 tan x| 42 ln | co s u | 4 co s4x tan x co s4x C .1 9. sinx co sxd x sin x co sxd co s x (1 u 2 ) 2 u 2 du (u 21 2 u u 6 )dx 21 u 3 312 u 5 5u 7 C7 (co s x ) 33(co s x )5(co s x )7C.2 0. sin 1 co s 2 x 3xdx dx21 (1 3 co s 2 x 3 co s 28x co s 3 2 x)dxx331sin 2

15、x (1 co s 4 x)dx (1 2 2 x )d sin 2 x816x31 63 11 61 1sin 2 x x 4 x sin 2 x sin 3 2 x C816 C .16 416 311 sin 3 x sin x 22 1. sin 2 x co s 4 xd x sin 2 2 x co s 2 xd x dx1 2 3 x sin 1 6442x 2 sin 3 x sin x d x1 1 co s 6 1 co s 2xco s 2 x co s 4 co s 2 x 16221 x 1 sin 2 x 1 sin 4 x 1 sin 6 x C.16 4412另

16、解sinx co sxd x 1 co s 2 x 1 co s 2 x 2dx221 (1 co s 282 x 2 co s 2 x )(1 co s 2 x ) dx 1 1 co s 282 x 2 co s 2 x co s 2 x co s 3 2 x 2 co s 22 x d x 1 1 co s 2 x co s 282 x co s 3 2 x d x 1 x 1 sin 2 x 1 (1 co s 4 x )dx 1 (1 2 2 x )d sin 2 x8 216161 11 11 1 x sin 2 x x sin 4 x sin 2x 3 2 x C8 216 41

17、6 31x 1 6sin 4 x 6 4dx1sin4 83 2 x C.xdu2 2. I .sin x 2 co s x 2 du u , x 2arctanu , dx 2 1 u 2I 1 u 2 2du du du 2u2 (1 u2 2 u 2 2 u 2u 2 u 11 2251 u51 u 2 u 2 251ln5u 1 u 1 2u 1 252522u 2u 5 12u 5 12 3. sin x co s xsin 2 x co s 4 xdx tan xd tan x udu (u tan x )tan12 x (1 tan 2du 2x ) 11u 2 (1 u 2 )

18、 1dv ( v u 2 )2u 2 (1 u 2 ) 12v (1 v ) 1 1 2dv1331 22223arctan32 v 1 C3 v 32 233dw133dw2 tan 2x 1 C .另 解 : I 1 d sin 2 x 1 ( w sin 2 x)2sin2 x (1 sin 2x )2w (1 w )2 1 2dw13333122 w 13arctan C 32 sin x 1 C .3(w )3 22dx124. (1 co t 2 x )d co t x co t x co t x C .1 x 2sin 41 x 22 5.1 1 dx x1 x 2dx arcs

19、inx C.x 12 6. I 1x 11 3x 1dx.x 1 u , x 1 u 6 , dx 6 u 5 du ,I 6 (1 u 3 )u 5 du1 u 2 u 5 u 8 du u66(1 u 266(uuu u 1 u 1 1 u 2)dx 111111 6 u7 u5 u 4 u3 u 2 u ln (1 u 2 ) arctanu C . 7543222 7.x 1 x 1 x x dx x 1 x 1 x 1 x 12x 1 x 1d x 2 x 22x 2 1dx 1 x 2 1 xx 2 x 2 11 ln ( x2x 2 1 ) C .3 ( x 1) 2 ( x 1

20、) 42 8. 3 ( x 1) 2 ( x 1) 4x 1dx.3( x 2 1) 3x dx.3( x 2 1) 3x 1x 1x 1x 1x 1x 1 ( x 1) u 3 , x 1 u 1 u 1 21 u 36 u 2 du,dx ,(1 u 3 )2I 6u 2 du(1 u 3 ) 2 6 udu 3 u1 C 3x 3x 12x 13 1 u 3 1 u 3 12(2 u 3 )2 u x 2 x 2 x 32 9.xd x 1 x 2 x 2 x 3xdx 1 x 2 x 2 x 32 x 1 1dx 1 2d ( x x 3) 1 dxx 2 x 2 x 3 x 1 22 21 1 2x 2 x 31x 2 x 3ln x x 2 x 3 C .22x3 0. I dx .(1 x 1 / 3 )1/ x(1 x 1 / 3 )1 / 2 u , x (u 1) 3 , dx 3( u 2 1) 2 (2 u ) du ,I 6 (u 2 1) 3 (u2u1)2 (u)du 6 (u 6 3u 4 3u 2 1)( u 2 u1)du