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⃝

f(x)

F(x)

•F′(x) = f(x)

•

•

•Æ

•

•

√−2 tan x dx =−1

2ln 2−2 tan x−2√−2 tan x

−arctan √−2 tan x−1

+1

2ln 2−2 tan x+ 2√−2 tan x

−arctan √−2 tan x+ 1,

√−2 tan x dx =

−[x]

2[x]−21−√2[x]

+2 1 + √2[x]

+1−√2[x] + [x]

−1 + √2[x] + [x] ,

= ln cos x+ ln(1 + √−2 tan x−tan x)

+ arctan 1 + tan x

√−2 tan x,

=−arctan √−2 tan x

1 + tan x+ arctanh √−2 tan x

1−tan x,

= arctan 1 + tan x

√−2 tan x+ arctanh 1−tan x

√−2 tan x.

f F

F′=f

dx

x=ln x ,

ln |x|,

(nπ, nπ +π/2) n∈Z

F′=f

√−2 tan x x =nπ +π/2

x=nπ

x= (n+

1/2)π

tan x=±1

x=nπ +π/2

x2+ 2

x4−3x2+ 4 dx = arctan x

2−x2,

= arctan 2x+√7+ arctan 2x−√7.

Æ

√2 tan x dx =√tan xcos xarccos (cos x−sin x)

√cos xsin x

−ln cos x+√2√tan xcos x+ sin x.

x→ −x

f(ax +b)dx

y=ax +b y

(m+n+ 1)

m+n+ 1 ̸= 0

T1= tan(c+dx)m m −1

m≥1

m+n+ 1 = 0

C= 0

tanm(c+dx)(a+btan(c+dx))n

∗(A+Btan(c+dx) + Ctan2(c+dx)) ,

a, b, c, d, A, B, C ∈Cm, n ∈R

m n

B=C= 0

a2+b2= 0

u= tan(c+dx)

u

u= tan((c+dx)/2)

tan(1 + i+x)∗

4−12 tan(1 + i+x) + 9 tan(1 + i+x)2

(2 −3 tan(1 + i+x))3/2dx .

m= 1 n=−3/2n≤ −1

Ab2−abB +

a2C= 0

−1

13 tan(1 + i+x)(−26 + 39 tan(1 + i+x))

2−3 tan(1 + i+x)dx .

tan(1 + i+x)2−3 tan(1 + i+x)dx .

dn T1Tn

2dx −→ Tn

2−dn Tn−1

2T3(b, −a, 0) dx

3 + 2 tan(1 + i+x)

2−3 tan(1 + i+x)dx + 22−3 tan(1 + i+x).

A2+B2̸= 0 a2+b2̸= 0

A+Btan(c+dx)

a+btan(c+dx)dx −→

1

2(A−Bi)1 + itan(c+dx)

a+btan(c+dx)dx+

1

2(A+Bi)1−itan(c+dx)

a+btan(c+dx)dx

A2+B2= 0 bA +aB ̸= 0

A+Btan(c+dx)

a+btan(c+dx)dx −→

−

2Barctanh √a+btan(c+dx)

√a+bA

B

da+bA

B

+ 22−3 tan(1 + i+x)

−√2−3iarctanh 2−3 tan(1 + i+x)

√2−3i

−√2 + 3iarctanh 2−3 tan(1 + i+x)

√2+3i

(sin(c+dx))−1

csc(c+dx) tan(c+dx)−1

cot(c+dx)

T1= tan(c+dx), T2=a+btan(c+dx),

T3(A, B, C ) = A+Btan(c+dx) + Ctan2(c+dx).

A, B, C ∈C

d(m+ 1) Tm

1Tn

2T3(A, B, C )dx =

AT m+1

1Tn

2+dTm+1

1Tn−1

2T3(ˆ

A, ˆ

B, ˆ

C)dx ,

ˆ

A=aB(m+ 1) −Abn ,

ˆ

B= (bB −aA +aC)(m+ 1) ,

ˆ

C=bC(m+ 1) −Ab(m+n+ 1) .

d(m+n+ 1) Tm

1Tn

2T3(A, B, C )dx =

CT m+1

1Tn

2+dTm

1Tn−1

2T3(ˆ

A, ˆ

B, ˆ

C)dx ,

ˆ

A=Aa(m+n+ 1) −C(m+ 1)a ,

ˆ

B= (aB +bA −bC)(m+n+ 1) ,

ˆ

C=aCn +bB(m+n+ 1) .

bd(n+ 1) a2+b2Tm

1Tn

2T3dx =

Ab2−abB +a2CTm

1Tn+1

2+dTm−1

1Tn+1

2ˆ

T3dx ,

ˆ

A=−Ab2−abB +a2Cm ,

ˆ

B=b(bB +aA −aC)(n+ 1) ,

ˆ

C= (m+n+ 1)(aB −Ab)b−ma2C+ (n+ 1)b2C .

ad(n+ 1) a2+b2Tm

1Tn

2T3dx =

−Ab2−abB +a2CTm+1

1Tn+1

2+dTm

1Tn+1

2ˆ

T3dx

ˆ

A=Aa2(n+ 1) + b2(m+n+ 2)

−a(bB −aC)(m+ 1) ,ˆ

B=a(aB −bA +bC)(n+ 1),

ˆ

C=Ab2−abB +a2C(m+n+ 2) .

bd(m+n+ 1) Tm

1Tn

2T3(A, B, C )dx =

CT m

1Tn+1

2−dTm−1

1Tn

2ˆ

T3dx ,

ˆ

A=aCm , ˆ

B=b(C−A)(m+n+ 1) ,

ˆ

C=aCm −bB(m+n+ 1) .

ad(m+ 1) Tm

1Tn

2T3(A, B, C )dx =

AT m+1

1Tn+1

2+dTm+1

1Tn

2ˆ

T3dx ,

ˆ

A=aB(m+ 1) −Ab(m+n+ 2) ,

ˆ

B=−a(A−C)(m+ 1) ,ˆ

C=−Ab(m+n+ 2) .

a2+b2= 0

T3

d(m+ 1) Tm

1Tn

2T3(A, B, 0) dx =

AaT m+1

1Tn−1

2−dTm+1

1Tn−1

2T3(ˆ

A, ˆ

B, 0) dx ,

ˆ

A=Ab(n−1) −(Ab +Ba)(m+ 1) ,

ˆ

B=Aa(m+n)−Bb(m+ 1) .

d(m+n)Tm

1Tn

2T3(A, B, 0) dx =

BbT m+1

1Tn−1

2+dTm

1Tn−1

2T3(ˆ

A, ˆ

B, 0) dx ,

ˆ

A=Aa(n+m)−Bb(m+ 1) ,

ˆ

B=Ba(n−1) + (Ab +Ba)(m+n).

2a2nd Tm

1Tn

2T3(A, B, 0) dx =

BbT m

1Tn

2+dTm−1

1Tn+1

2T3(ˆ

A, ˆ

B, 0) dx ,

ˆ

A= (Ab −Ba)m , ˆ

B=Bb(m−n) + Aa(m+n).

2a2nd Tm

1Tn

2T3(A, B, 0) dx =

−a(aA +bB)Tm+1

1Tn

2+dTm

1Tn+1

2ˆ

T3(ˆ

A, ˆ

B, 0) dx ,

ˆ

A=bB(m+ 1) + aA(m+ 2n+ 1) ,

ˆ

B= (aB −Ab)(m+n+ 1) .

ad(m+n)Tm

1Tn

2T3(A, B, 0) dx =

aBT m

1Tn

2+dTm−1

1Tn

2ˆ

T3(ˆ

A, ˆ

B, 0) dx ,

ˆ

A=−aBm , ˆ

B=Aam + (Aa −Bb)n .

ad(m+ 1) Tm

1Tn

2T3(A, B, 0) dx =

aAT m+1

1Tn

2+dTm+1

1Tn

2ˆ

T3(ˆ

A, ˆ

B, 0) dx ,

ˆ

A=Abn −Ba(m+ 1) ,ˆ

B=Aa(m+n+ 1) .