japan today's calculation of integral 2005 184

5/21/2018 Japan Today's Calculation of Integral 2005 184

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JapanTodays Calculation Of Integral

2005

1 Calculate the following indefinite integral.

[1]

e2x

(ex+1)2dx

[2]

sin x cos3xdx

[3]

sin2x sin3xdx

[4]

dx4x212x+9

[5]

cos4 xdx

2 Calculate the following indefinite integrals.

[1]

cos

2x 3

dx

[2] dx

cos2(3x+4)

[3]

(x 1) 3x 2dx[4]

x 3x2+1dx[5]

dx1xdx


[1]

sin x sin2xdx

[2]

e2x

ex1 dx

[3] tan2 xcos2 x dx[4] ex+ex

exex dx

[5]

ex

ex+1 dx


[1]

x5xdx

[2]

sinx cos2 x1+cosx dx

[3]

(sin x+ cos x)2dx

[4] xcos2 xx cos2 x

dx

[5]

(sin x+ sin 2x)2dx


[1]

(4 5tan x)cos xdx[2]

dx3

(13x)2 dx

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2/14


2005

[3]

x3

4 x2dx[4]

ex sin

x+ 4

dx

[5]

(3x 4)2dx


[1]

sin x cos3 xdx

[2]

dx(1+

x)x

dx

[3] x2

x3 + 1dx

[4] e2x3ex

ex dx

[5]

(1 x2)exdx


[1]

x(

x+ 1)2dx

[2]

(ex + 2ex+1 3ex+2)dx[3]

(sin2 x+ cos x)sin xdx

[4]

x

2 xdx[5] x ln xdx


[1]

x(x2 + 3)2dx

[2]

ln(x+ 2)dx

[3]

x cos xdx

[4]

dx(x+2)2

dx

[5]

x1x22x+3 dx


[1]

(x2

+ 4x 3)

2

(x

+ 2)dx

[2]

ln xx(lnx+1) dx

[3] sin (log2x)

x dx

[4]

dxsinx cos2 x

[5]

1 3x dx


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3/14


2005


[1]

(2x+ 1)

x+ 2 dx

[2]

1+cosxx+sinx dx

[3]

sin5 x cos3 x dx

[4] (x3)2

x4 dx

[5]

dxtanx dx


[1]

6x+13x2+x+4

dx

[2]

ex

ex+eaxdx

[3] (x+1)3

x dx

[4]

x ln(x2 1)dx[5] 2(x+2)

x2+4x+1dx


[1] dx1+cosx

[2]

xx2 1dx[3]

ax

2 dx (a >0, a= 1)[4]

sin3 x1+cosxdx

[5]

e4x sin2xdx

13 Calculate the following integarls.

[1]

x cos2 xdx

[2]

x1(3x1)2 dx

[3] x3

(2

x2)4

dx

[4]

14x

+ 12x

dx

[5]

(ln x)2dx


[1]

sinx cosx1+sin2 x

dx

[2]

x log10xdx


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4/14


2005

[3]

x2x1 dx

[4]

(x2 + 1) ln xdx

[5]

ex cos xdx


[1] (x21)2

x4 dx

[2]

e3xex+1

dx

[3] sin2x cos3xdx[4]

x ln(x+ 1)dx

[5]

x(x+3)2

dx


[1]

sin(ln x)dx

[2] x+sin2 xx sin2 x

dx

[3]

x3

x2+1dx

[4] x2

2x1 dx

[5] x+cos 2x+1

x cos2 x dx


[1]

dxexex

[2]

eax cos2xdx(a= 0)[3]

(3x 2)2dx[4] x4+2x3+3x2+1

(x+2)5 dx

[5]

dx1cosxdx

18 Calculate the following indefinite integrals.[1]

(sin x+ cos x)4dx

[2]

e2x

ex+1 dx

[3]

sin4 xdx

[4]

sin6x cos2xdx

[5]

x2(x+1)3

dx


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5/14


2005


[1]

tan3 xdx

[2]

amx+ndx(a >0, a= 1, mn= 0)[3]

cos5 xdx

[4]

sin2 x cos3 xdx

[5]

dxsinx


[1]

ln(x2 1)dx[2]

1ex+1 dx

[3]

(ax2 +bx+c)emxdx(abcm= 0)[4]

tan x+ 1tan x2

dx

[5]

1 sin xdx

21 [1] Tokyo Univ. of Science:

lnx(x+1)2

dx

[2] Saitama Univ.:

53sinx+4cosxdx

[3] Yokohama City Univ.:

3

1

1

x2

+1

dx

[4] Daido Institute of Technology:

2

0sin3 x

sinx+cosxdx

[5] Gunma Univ.:3

4

0 {(1 +x)sin x+ (1 x)cos x}dx

22 Evaluate

10

(1 x2)ndx(n= 0, 1, 2, )

23 Evaluate

lima

20

a0

(cos x) ln(cos x) dx

0 a <

2

24 Find the minimum value of

0(x y)2(sin x)| cos x|dx.


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6/14


2005

25 Let|a|< 2 .Evaluate

2

0

dx

{sin(a+x) + cos x}2

26 Evaluate

eeee

eee

dx

x ln x ln(ln x) ln{ln(ln x)}27 Let f(x) =t sin x+ (1 t)cos x(0 t 1).

Find the maximum and minimum value of the following P(t).

P(t) =

2

0exf(x)dx

2

0exf(x)dx

28 Evaluate

40

x cos5x

cos x dx

29 Let a be a real number. Evaluate

3+a+a

|x a | sinx

2

dx

30 A sequence{an}is defined by an=1

0 x3(1 x)ndx(n= 1, 2, 3. )

Find the constant number c such that

n=1(n+c)(an an+1) = 1331 Evaluate

limn

0

x2| sin nx|dx

32 Evaluate

10

ex+ex+ee

x

+eeex

dx


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7/14


2005

33 Evaluate

0 ln 2

dx

cos2 hx 1 2a tanh x+a2 (a >0)

34 Let p be a constant number such that 0 < p


8/14


2005

40 Evaluate

10

x2005ex2

dx

41 Evaluate

a0

2ax x2 dx(a >0)

42 Let 0 < t < 2 .

Evaluate

limt

2

t0

tan

cos ln(cos )d

43 Evaluate

2

0cos2004 x cos 2004x dx

44 Evaluate

2

0

sin 2005x

sin x dx

45 Find the functionf(x) which satisfies the following integral equation.

f(x) =

x0

t(sin t cos t)dt+

2

0etf(t)dt

46 Find the minimum value of1

0 |t

x

|t+1 dt47 Find the condition ofa, b for which the function f(x) (0 x2) satisfying the following

equality can be determined uniquely,then determine f(x), assuming that f(x) is a continuousfunction at 0x2.

f(x) = a

2

20

sin(x+y)f(y)dy+ b

2

20

cos(x y)f(y)dy+ sin x+ cos x




9/14


2005

48 Evaluate

limn

0

sin2 nx

sin x dx

nk=1

1

k

49 For x0, Prove that x0(t t2)sin2002 t < 12004200550 Let a, b be real numbers such that a < b. Evaluate

limba

ba

ln |1 + (x a)(b x)|dx(b a)3

.

51 A function f(x) satisfies f(x) = fcx

for some real number c(>1) and all positive number

x.

Ifc

1f(x)x

dx= 3, evaluatec

1f(x)x

dx

52 Evaluate

limn

nk=1

1n+k1

53 Find the maximum value of the following integral.

0

ex sin tx dx

54 evaluate

0

1 1 +x

1 xdx

55 Evaluate

limnn

10

(1 +x)n1ex2

dx (n= 1, 2, )




10/14


2005

56 Evaluate

limn

nk=1

[

2n2 k2 ]n2

[x] is the greatest integerx.57 Find the value ofn N satisfying the following inequality.

0

x2

sin nx dx < 99

2

100n

58 Let f(x) = ex

ex+1

Prove the following equation.

ba

f(x)dx+

f(b)f(a)

f1(x)dx= bf(b) af(a)

59 Evaluate

(cos2x)(cos 22x) (cos22006x)dx

60 Let an=

2

0 sin2t(1 sin t)n1

2 dt(n= 1, 2, )Evaluate

n=1

(n+ 1)(an an+1)

61 Evaluate

2004k=0

11

1 x2k+ 1 x dx

62 For a >1, let f(a) = 121

0|axn 1|dx + 12 (n= 1, 2, ) and let bn be the minimum value off(a) at a >1.

Evaluate


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11/14


2005

limm bm bm+1 b2m (m= 1, 2, 3, )

63 For a positive number x, let f(x) = limnn

k=1

cos 2k+12n x cos 2k12n xEvaluate

limx

f(x)

x

64 Let f(t) be the cubic polynomial for t such that cos 3x= f(cos x) holds for all real numberx. Evaluate

10{f(t)}2

1 t2dt

65 Let a >0. Find the minimum value ofa1

1 xa

1 +x dx

66 Find the minimum value of

2

0 | cos x a| sin x dx

67 Evaluate

2005

1002

0

dx10022 x2 + 10032 x2 +

1003

1002

10032 x2dx 1

0

1 x2dx

68 Find the minimum value ofe

1

ln x ax

dx(0ae)69 Let f1(x) =x, fn(x) =x+

114

0 xfn1(t)cos

3 t dt(n2).Find limn fn(x)

70 Find the number of root for

2

0 ex cos(x+a) dx= 0 at 0

a


12/14


2005

74 p, qsatisfies px+qln xat axb (0 < a < b).Find the value of p, q for which the following definite integral is minimized and then theminimum value.

ba

(px+q ln x)dx

75 A function f() satisfies the following conditions (a), (b).

(a) f()

0

(b)

0 f()sin d = 1

Prove the following inequality.

0

f()sin n dn (n= 1, 2, )

76 The function fn(x) (n= 1, 2, ) is defined as follows.

f1(x) =x, fn+1(x) = 2xn+1 xn +1

2

10

fn(t) dt (n= 1, 2, )

Evaluate

limn fn

1 +

1

2n

77 Find the area of the part enclosed by the following curve.

x2 + 2axy+y2 = 1 (1< a


13/14


2005

80 Let Sbe the domain surrounded by the two curves C1 : y = ax2, C2 : y =ax2 + 2abx for

constant positive numbers a, b. Let Vx be the volume of the solid formed by the revolution ofSabout the axis ofx, Vy be the volume of the solid formed by the revolution ofS

about the axis ofy. Find the ratio of VxVy

.

81 Prove the following inequality.

1

12( 6 + 2

3)

4

6

ln(1 + cos 2x)dx14

(2

3)

82 Let 0 < a < b.Prove the following inequaliy.

1

b a ba

ln

b

x

2dx


14/14


2005

87 Find the minimum value ofa(0 < a

japan today's calculation of integral 2005 184

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