2007-08+

Z

ln(1 +

p

x)dx:

3.

3

Найти yx0 и yxx00 ; если

p

p

p

p

x(t) = ch t + 3 t;

y(t) = ch t 3 t:

4.

4

Найти y(n) при n 2; если

p

y = (x

2

+ 3x + 1) 2x + 1:

p

x

x!0

1 + x

1 + 2 cos

arcsin x

(1 ch x)2

lim

x

x

:

7. 5 Вычислить предел

lim

ex x2 ch x2 ln(1 + x)

:

x!0 tg(sh x) arctg(sh x)

2а.

1

3

2

1

2x+1

3 ln

3x

1 +

2 ln(x

+ x + 1)

p

arctg

p

+ C

j

j

p

x

p

3

3

x)

+

2б. (x 1) ln(1 +

2

x + C

p

p

3. yx0 = sh p

t

3

; yxx00 =

6 ch t

3

(sh pt+3)

sh

t+3

4. y(n) = (x2 + 3x + 1)n!C1n=2 2n(2x + 1)1=2 n+

+n(2x + 3)(n

1)!Cn 1 2n 1(2x + 1)3=2 n+

n(n 1)

1=2

+

2 (n

2)!Cn 2

2n 2(2x + 1)5=2 n

2

1=2

5.

x 2

n 1 (x 2)2k+1

2n

y =

ln 4

+ o (x

2)

2

2k+1

k

2

=1

143

6. exp

kP

f(x) =

(1 + x

x2 + o(x3))

x

x2 + 67x3 + o(x3) = 1

67x3

+ o(x3)

4

g(x) =

x3+o(x3)

7.

47

1x2

5x3

o x3

x3

1x2

1x3

o x3

x

1 +

x

)

o

))

x

+

) =

f( ) =

2

6

+ (

(1+ (

2

3

+ (

= 67x3 + o(x3)

x!0

1 + x

+ p1 + x

1

lim

cos x

arctg x

sin x x

2a.

1

1

2

3

2x+1

4 ln j4x + 1j +

2 ln(x

+ x + 2) + p

arctg

p

+ C

7

7

1+ t

2б. (x

x + p1

x + C

2) arctg p1

p

1+p

t

3. y0

=

1+e

;

y00

=

2e

p

3

x

1 e1+ t

xx

(1 e1+pt)

+n (2x) 2n 1

sin

2x +

(n 1)

+

2

n(n 1)

n

2 sin 2x +

(n 2)

+

2

2n 21

k

2k+1

2

2n

( 1)

2 (x+1)

5. y = 1 + k=0

+ o (x + 1)

3k+1

11

P

6. e

4

f(x) = 1 x + 21 x2 21 x3 + o(x3) + x

21 x2 +

1

x3 + o(x3) = 1

24

2411 x3 + o(x3)

1

g(x) =

61 x3+o(x3)

7. 45

1 + x + 23 x2 + 67 x3 + o(x3) (1+

23 x2+o(x3)) x +

31 x3 + o(x3)

f(x) =

=

= 65 x3 + o(x3)

Z

p

ln( x 1)dx:

3.

3

Найти yx0 и yxx00 ; если

p

p

p

p

x(t) = 2 t + ln(1 +

t);

y(t) = 2 t ln(1 +

t):

4.

4

Найти y(n) при n 2; если

p

x2

y =

:

x!0

1 + x

+ p1 2x

x

lim

cos 2x

arcsin x

(1 cos x)2

lim arctg(xe 2x) 41 ln(1 + 4x)

x!0

cos(x + x2) + ch x 2

2a.

1

j

p j

3

2

7

2x+3

2 ln

x

p

p

arctg

p

+ C

3

3

2x + 1 + 2 ln(x + 3x + 3)

2б. (x 1) ln(

1) 2 p

+ C

x

x

3. yx0

1+2p

00

4(1+ t)

t

=

p

;

yxx =

3

t

(3+2pt)

3+2

4.

y(n) = x2

n!Cn

4n(4x + 3) 31 n+

1=3

4n 1(4x + 3)

1

+n(2x)(n

1)!Cn 1

3 n+1+

1

n(n 1)

1=3

+

2(n

2)!Cn 2

4n 2(4x + 3) 3 n+2

2

1=3

ln 3

n 1

( 1)k 1(x 3)2k+1

2n

6. e 3

(x

P

3k+1k

5. y = 3

3) k=1

+ o (x 3)

32

f(x) = 1 x x2 + x3 + o(x3) x + x2

+ 35x3 + o(x3) = 1 +

38x3 + o(x3)

4

g(x) =

x3+o(x3)

7.

113

35x3 + o(x3) x 2x2

163 x3 + o(x3) = =

f(x) = x 2x2 +

+

113 x3 + o(x3)

arctg(

x + 3)dx:

3.

3

Найти yx0 и yxx00 ; если

p

p

x(t) = p

arcsin

t

;

y(t) = p

+ arcsin

t

:

t

t

2

2

4.

4

Найти y(n) при n 2; если

x

1

x!0

1 + x2

2

lim

e

1p1 + x arctg x

sh x x

lim

arcsin(ln(1 + x)) + e x

1

3

x!0

ex+x sin x ch x

Ответы к варианту МФТИ-74.

2a. 21 ln j2x 1j + 25 ln(x2 2x + 2) + 4 arctg(x 1) + C

2б. (x + 4) arctg p

p

+ C

x + 3

x + 3

p

p

3.

y0

=

4 t

+1

;

y00

=

2 t

3

x

p4 t 1

xx

(p4 t 1)

4. y

(n)

= (2x

2

+ 3x)

3n

( 1)n 1(n 1)!

+

ln 3

(3x+2)n

n 1

( 1)n 2(n 2)!

+n (4x + 3) 3ln 3

(3x+2)n 1

+

n(n 1)

3n 2

( 1)n 3(n 3)!

+

2

4

ln 3

(3x+2)n 2

n 1

3 ( 1)k (x 2)2k+1

2n

6. e

P

8

2k+1

5. y = 1 + k=0

+ o (x 2)

13

f(x) = 1 + 21x + 41x2 +

1

x3 + o(x3)

21x +

41x2 4811x3 + o(x3) =

24

1 + 48x

+ o(

) 1

13

3

x3

g(x) =

1 x3

+o(x3)

7.

41

6

f(x) = x 21x2 + 21x3 + o(x3) + 1 x + 21x2 61x3 + o(x3) 1 =

31x3 + o(x3)

+ o(x3) 1 +

g(x) = 1 + x + 21x2 + 67x3 + o(x3) x

61x3

21x2 + o(x3) =

= 34x3 + o(x3)