不定积分换元法例题1

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【第一换元法例题】

1、9999(57)(57)(571

1(57)(57)55

)(57)dx d x d x dx x x x x +=+?=+?=+?++???? 110091(57)(57)(57)10111(57)5550

d C x x x x C =?=?+=+++++? 【注】1(57)'5,(57)5,(57)5

x d x dx dx d x +=+==+??

2、

1ln ln ln ln dx d x x x dx x x x =?=????

221(l 1ln ln (ln )2n )2

x x x d C x C =?=+=+? 【注】111(ln )',(ln ),(ln )x d x dx dx d x x x x ===??

3（1）sin tan cos co si s cos cos n cos cos xdx d x xdx dx x d x x x x x --====?????

cos ln |cos |c ln |co s |o s x x d C x C x

=-=-+=-+? 【注】(cos )'sin ,(cos )sin ,sin (cos )x x d x xdx xdx d x =-=-=-??

3（2）cos cos cot sin sin sin sin xdx x xdx dx d x x x x ===????

sin ln |si ln |sin |n |sin x x d C x C x

==+=+? 【注】(sin )'cos ,(sin )cos ,cos (sin )x x d x xdx xdx d x ==?=?

4（1）1()11d dx a x a x a d x x a x =?=?++++???

ln |1(|)ln ||d C a x a x a x a x

C ++=?=+=+++? 【注】()'1,(),()a x d a x dx dx d a x +=+==+??

4（2）1()11d dx x a x x x d a a x a =?=?----???

ln |1(|)ln ||d C x a x a x a x a

C --=?=+=--+? 【注】()'1,(),()x a d x a dx dx d x a -=-==-??

4（3）22221111111212x a a x a dx dx x a x a dx dx a a a x dx x ??- ?--+???=-+?==- ?-??

????? ()11ln ||ln ||ln 22x a x a x a C C a a x a -=--++=++

5（1）2sec ()sec tan sec sec tan sec tan sec sec tan x x x x x dx x x x xdx dx x x

+==?+++??? tan sec tan sec sec ()()ln |sec tan |se tan c tan d x x x x x x

d x x C x x +===+++++?

? 5（2）222sec cos c os cos 1sin xdx dx dx x x x x ====-????? 2sin si 1111sin 111sin ln ln 1n sin 2112sin 121s sin sin in d x x x x x x d C C x x

x --??==-?=+=+ ?--+++???? 6（1）2csc ()csc cot csc csc cot csc cot csc csc cot x x x x x dx x x x xdx dx x x

+==?+++??? ()()ln |csc cot |csc c cot csc csc cot csc o ot t c d d x x x x x x

x x C x x --+=-==+-+++?? 6（2）2csc ()csc cot csc csc cot csc cot csc csc cot x x x x x dx x x x xdx dx x x

==?----??? ()(cot csc csc co )ln |csc t csc co cot |c t sc cot d x x x x d x x x x x C x -+-=---==+?

? 7（1）22arcsin 11dx x C x x ==+--

7（2）2222222arcsin 111d x dx C a

a x d x x x a x x x a a a a =====+--????--- ? ??????? ?????? ???? ????? 8（1）221arctan 11dx dx x C x x ==+++?? 8（2）222222221111arctan 111d dx x dx C a x a x a a a x x x d dx x a x a a a a a a ???? ?=====+++??????++???? ?+?? ? ?????????

??????????，（0a >）

9（1）352525s sin cos sin cos sin i c s o c n o s xd x xdx x x x x x d x =?-?=???

86257

5cos cos (1cos )cos cos (cos cos )cos 86x x x x d x x x d x C =--??=-?=-+?? 9（2）353434

c sin cos sin cos sin cos os sin x x xdx x x x dx

d x x =?=???? 468322357sin sin sin sin (1sin )sin (sin 2sin sin )sin 438x x x x x d x x x x d x C =-?=-+?=-++??

10（1）

1ln 111l l n ln ln l ln n n ln dx d x C x x x x dx d x x x x =?=?=?=+?????

10（2）222211111ln ln ln ln ln n ln l dx d C x x x x d x x x x d x x ?=?=?=?=-+????

11（1）242424222222()arctan(21)222)

121122(xdx d x C x x x x x x x x dx x dx ====+++++++++++???? 11（2）2242422422121()2521112252524()

xdx d x xdx d x x x x x x x x +===++++++++???? 2222222121(1)111arctan()8442111122x d d x x C x x ??+ ?++??===+????++++ ? ?????

??

12、s 222x dx x x dx dx x x x x

x =?=?=??? 2C x C x x x ==-=-?

13、22221122212

2x x x x e dx e d x d e x C e =

==+??? 14、 43333

co sin sin cos sin sin s sin i 4sin s n x x xdx x x d C dx x x x d x =?=?=?=+????

15、100

(25)x dx +?10010010011(25)(25)2(25)(25)(25)2

dx d x x x x d x =+?=+++?+?=??? 1001100111(25)(25)(25)101111(25)22202x x x d C x C =?=?+=+++++?

16、2222222111sin sin s 2in sin cos 22x x x x x dx x xdx dx x d C =?=

?=?=-+????

17、ln 1ln 1ln 1ln 1ln 1ln dx d d x x x x x x x x ===++++?

31221ln ln ln 1ln 1ln (1ln )(1ln )1ln 2(1ln )2(1ln )3

x d x d x x x d x d x x

x x C =+-+=++-++=+-++?

?

18、arctan arctan arctan arc arct 2tan 2an arcta 11arct 1n an x x x x x e dx e e e d e C x dx d x x

x +=?=?=?=++???? 19、2222

22(11111)22x x x x d x xd dx x ===------ 2221(1)21d x x x C -=-=--

20、

333si cos cos cos n cos x dx d x x x x =-=?3221cos cos 2cos x C x d x --=-=+?

21、111()ln(22222)2x x x x x x x x x e dx d e e dx d e C e e e e

e =?=?==+++++++????

22、23222ln ln ln l 1ln ln ln n 3

x x dx x x x x d C x dx d x x =?=?=?=+????

23、

222222

122(1)2(1)(2)(1)C x x x x x ====+---++-+- 24、22222

1()21777()1121()()()()22424d x dx x x x x x d x dx -===-+-+-+-+-??? 22

1()217777()21722()2

d x C C x x -==-+=-++?

25、计算2222sin cos dx a x b x +?，22a b ≠

【分析】因为：22222222(sin cos )'2sin cos 2cos (sin )2()sin cos a x b x a x x b x x a b x x +=+-=- 所以：222222(sin cos )2()sin cos d a x b x a b x xdx +=-

2222221sin cos (sin cos )2()

x xdx d a x b x a b =?+- 【解答】2222222222222222221sin cos sin c os 2sin cos a x b x a x b x dx a b a x b x ==+++- 22222222222222222si 11sin cos n cos a a x b x x b x C a b a b

+==+--

【不定积分的第二类换元法】

已知()()f t dt F t C =+?

求()(())()(())'()g x dx g t d t g t t dt ????==?

??

【做变换，令()x t ?=，再求微分】 ()()f t dt F t C ==+? 【求积分】 1(())F x C ?-=+ 【变量还原，1()t x ?-=】

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【第二换元法例题】

1、22sin sin 2si 2n x t x t x t t dx tdt t t dt tdt x

=?=?=========????令 2cos t x t C x C =========-+-+变量还原

2（1）22111221112111x t x t dt td t dx dt dt t t t t x t =========???=?==- ?+++++???????令

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