Iracionālo funkciju

integrēšana

I. paņēmiens

dxdcx

baxxRI n

,

tdcx

baxn

ntdcx

bax

n

n

cta

bdtx

dtcta

bdtdx

n

n

ntdcx

bax

ntdcxbax

nn dtcxtbax

bdtcxtax nn

bdtctax nn

n

n

cta

bdtx

dxx

x

12 tx 12212 tx 12 2 tx

2

12 t

x

tdtdx

1

2

21 2

2

2 t

dtttdt

t

t

dtt

dtdtt

t

1

122

1

112

22

2

Cx

xx

Ct

tt

112

112ln122

1

1ln2

II. paņēmiens

Caxxaaxxdxax

Caxxaaxxdxax

Ca

xaxaxdxxa

2222222

2222222

22222

ln2

1

ln2

1

arcsin2

1

dxxx 1342

dxx 92 2

Cxxxxxx

1342ln

2

91342

2

1 22

III. paņēmiens

dx

cbxax

bMaN

bax

a

Mdxcbxax

NMx22

22

2

dxbaxcbxaxd 22

cbxax

dx

a

bMN

cbxax

cbxaxd

a

M22

2

22

III. paņēmiens pirmais gadījums

22

2

22

2 XA

dx

a

bMNcbxax

a

M

CA

X

a

bMNcbxax

a

M

arcsin

22

22

III. paņēmiens otrais gadījums

22

2

22

2 AX

dx

a

bMNcbxax

a

M

CAXxa

bMNcbxax

a

M

222 ln

22

2

dx

xx

x228

32

dxxxxd 2228 2

dx

xx

x228

122

22

2

19

1

28

)28(

x

xddx

xx

xxd

Cx

xx

3

1arcsin282 2

IV. paņēmiens

cbxax

dxxPn2

cbxax

dxkcbxaxxQn 2

21

172172

172

23522

2

2

2

xx

dxkxxBAxdx

xx

xx

Atvasina vienādojuma abas puses

172172

1172

172

235222

2

2

2

xx

k

xx

xBAxxxA

xx

xx

Atbrīvojas no saknes

kBBxAxAxxxAxx 222 1722352

kBBxAxAxAAxAxxx 222 1722352

2317

52

2

kBA

BAA

AA

2317

53

22

kBA

BA

A

2317

53

1

kB

B

A

23217

2

1

k

B

A

4

2

1

k

B

A

17241722

172

23522

2

2

2

xx

dxxxxdx

xx

xx

161

141722

2

2

x

xdxxx

Cxxxxxx 1721ln41722 22

V. paņēmiens

ctgtmuvaitgtmudumuuRI

t

muvai

t

mudumuuRI

tmuvaitmuduumuRI

223

222

221

,

cossin,

cossin,

324 x

dx tx sin2

3232 sin244 tx

3232 cos4sin44 tt

tt 33 cos8cos2

tdtdx cos2

2sin

xt

tt 2sin1cos 2

21cos

x

t

242

1cos xt

t

dt

t

tdt

x

dx2332 cos4

1

cos8

cos2

4

Cx

x

Ct

tCtgt

2421

24

1

cos

sin

4

1

4

1

Cx

x

244

1