Клевчихин - Матан II семестр

0

x

0

x

y = arctg x

y = arcctg x1

arcctg 1

0

x

1

1 x2

0

arcctg 1

1

1

x

2

x

a; b

arcctg 1

1

1 x

2

x

arcctg

1

arctg x,

x

0;

x

π arctg x,

x

0.

0;

arctg x

arcctg 1

; 0

x

F1

F2

f

a; b

C

x a; b

F1 x

F2 x C.

a; b

Φ x

F1 x

F2 x

x1

x2

a; b

Φ x1

Φ x2

Φ ξ x1

x2 .

ξ

a; b

Φ ξ

F1 ξ

F2 ξ

f ξ

f ξ

0

x1

x2

a; b Φ x1

Φ x2

0

Φ x

const

f

a; b

a; b

a; b

a; b

F

F

a; b

a; b a; b a; b

F

x

f x

x

a; b

a; b a; b a; b

F x

x

F x sgn x

x

0

F x

x

sgn x

a; b

xαdx

xα

1

C; α

1

dx

ln

x

C

α

1

x

ax dx

ax

C

ex dx

ex

C

ln a

sin x dx

cos x

C

cos x dx

sin x

C

dx

tg x

C

dx

ctg x

C

cos2 x

sin2 x

sh x dx

ch x

C

ch x dx

sh x

C

dx

th x

C

dx

cth x

C

ch2 x

sh2 x

dx

dx

ln x

x2

1 C

arcsin x

C

x2

1

1

x2

1

1

dx

dx

x

arctg x

C

ln

C

1 x2

1 x2

2

1

x

d

f x dx

f x dx

f x dx

f x .

ex sin x

ex cos x dx ex sin x ex cos x

ex sin x dx ex sin x ex cos x I.

u

cos x du

sin x dx

dv

ex dx

v ex

x

sin x

x

cos x I,

1

x

sin x

cos x .

I e

e

I

e

2

cos x

1

sin x

sin x

cos x

1

cos x

dx

dx

dx

sin x

sin x

sin2 x

sin x

u

1

du

cos x dx

sin x

sin2 x

dv

cos x dx v

sin x

cos x

dx

1

cos x

dx

sin x

sin x

f x dx F x

C

f ϕ t ϕ t dt F ϕ t

C.

F x

f x

F ϕ t

C

F ϕ t ϕ t f ϕ t ϕ t .

f ϕ t dϕ t

f x dx F x C F ϕ t

C.

ϕ t

x

x

ϕ t

x dx

x2

a2

x dx

dx2

d x2

a2

dt

t

C x2

a2

C.

x2 a2

2 x2

a2

2 x2

a2

2 t

tg x dx

tg x dx

sin x

dx

d cos x

dt

ln t C

ln

cos x

C.

cos x

t

cos x

arccos x 3 1

dx

dx

x dx

sh x

x4 6x2

5

1 x2

x3

ex dx

dx

x2 1 dx

x ln x ln ln x

4

e2x

ln2 x

arctg2 x

dx

dx

dx

x

1

x2

3 cos2 x

4 sin2 x

2 e3x dx

x

3

5x

arctg

4x 1 dx

dx

4x2

4x

3

A

A

a

x

a

A

n

1

x

a n

M x N

M N

p q

x2

px

q

D p2

4q 0

D p2

4q 0

M x N

D p

2

4q 0 n 1

x2

px q n

A

dx A ln x a

C.

x

a

A

A

C

n 1

dx

x a n

1 n x a n 1

M x N

M x

p

N

M p

dx

2

2

dx

x2

px

q

x

p 2

q

p

2

2

4

x

p

d x

p

dx

M

2

2

N

M p

x

p

2

q

p2

2

x

p 2

q

p2

2

4

2

4

d

x

p

2

q

p2

M p

d x

p

M

2

4

N

2

2

x

p

2

q

p2

2

x

p

2

q

p2

2

4

2

4