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Calculate the following indeﬁnite integral.

[1]

[2] sin x cos 3xdx

[3] sin 2x sin 3xdx

[4]

[5] cos 4 xdx

e 2x

(e x +1) 2 dx

dx

4x 2 12x+9

Calculate the following indeﬁnite integrals.

[1] cos 2x π dx

[2]

[3] (x 1) x 2dx

[4] x · 3 x 2 +1 dx

[5]

3

dx

cos 2 (3x+4)

3

dx

1x dx

Calculate the following indeﬁnite integrals.

[1] sin x sin 2xdx

[2]

[3]

[4]

[5]

1 dx

e x

tan 2 x cos 2 x

e x +e x e x e x

dx

dx

+1 dx

e x

e

x

e

2x

Calculate the following indeﬁnite integrals.

[1]

x

5x dx

sin x cos 2 x 1+cos x

[2]

[3] (sin x + cos x) 2 dx

dx

[4] xcos 2 x x cos 2 x

dx

[5] (sin x + sin 2x) 2 dx

Calculate the following indeﬁnite integrals.

[1] (4 5 tan x) cos xdx

[2]

dx

(13x) 2 dx

3

[3] x 3 4 x 2 dx

[4] e x sin x + π dx

[5] (3x 4) 2 dx

4

Calculate the following indeﬁnite integrals.

[1] sin x cos 3 xdx

[2]

[3] x 2 x 3 + 1dx

[4] e 2x 3e x

(1+ x) x dx

e

x

dx

dx

[5] (1 x 2 )e x dx

Calculate the following indeﬁnite integrals.

[1] x( x + 1) 2 dx

[2] (e x + 2e x+1 3e x+2 )dx

[3] (sin 2 x + cos x) sin xdx

[4] x 2 xdx

[5] x ln xdx

Calculate the following indeﬁnite integrals.

[1] x(x 2 + 3) 2 dx [2] ln(x + 2)dx [3] x cos xdx

[4]

[5]

dx

(x+2) 2 dx

x1

x 2 2x+3 dx

Calculate the following indeﬁnite integrals.

[1] (x 2 + 4x 3) 2 (x + 2)dx

[2]

ln x x(ln x+1) dx

sin (π log 2 x)

x

dx

sin x cos 2 x

[3]

[4]

[5] 1 3x dx

dx

Calculate the following indeﬁnite integrals.

[1] (2x + 1) x + 2 dx

[2]

[3] sin 5 x cos 3 x dx

1+cos x

x+sin x

dx

[4] (x3) 2

[5]

dx

x

dx

4

tan

x dx

Calculate the following indeﬁnite integrals.

[1]

[2]

6x+1

3x 2 +x+4 dx

e

x

e x +e

ax dx

[3] ( x+1) 3

x

dx

[4] x ln(x 2 1)dx

[5]

2(x+2)

x 2 +4x+1 dx

Calculate the following indeﬁnite integrals.

[1]

[2] x x 2 1dx

[3] a x dx

[4]

[5] e 4x sin 2xdx

dx

1+cos x

2

sin 3 x

1+cos

(a > 0, a

= 1)

x dx

Calculate the following integarls.

[1] x cos 2 xdx

[2]

x1

(3x1) 2 dx

[3]

x

3

(2x

2 ) 4

dx

[4]

[5] (ln x) 2 dx

1

4 x +

2x dx

1

Calculate the following indeﬁnite integrals.

[1]

[2] x log 10 xdx

sin x cos x 1+sin 2 x

dx

[3]

[4] (x 2 + 1) ln xdx

[5] e x cos xdx

x

2x1 dx

Calculate the following indeﬁnite integrals.

[1] (x 2 1) 2

x

4

dx

[2]

[3] sin 2x cos 3xdx

[4] x ln(x + 1)dx

[5]

e

3x

e x

+1 dx

x

(x+3) 2 dx

Calculate the following indeﬁnite integrals.

[1] sin(ln x)dx

[2] x+sin 2 x x sin 2 x

dx

[3]

[4]

+1 dx

x

2

x

3

2x1 dx

x

2

[5] x+cos 2x+1 x cos 2 x

dx

Calculate the following indeﬁnite integrals.

[1]

[2] e ax cos 2xdx (a

[3] (3 x 2) 2 dx

[4] x 4 +2x 3 +3x 2 +1 (x+2) 5

[5]

dx

e x e x

= 0)

dx

dx

1cos x dx

Calculate the following indeﬁnite integrals.

[1] (sin x + cos x) 4 dx

[2]

[3] sin 4 xdx

[4] sin 6x cos 2xdx

[5]

+1 dx

e

x

e

2x

x 2

(x+1) 3 dx

Calculate the following indeﬁnite integrals.

[1] tan 3 xdx

[2] a mx+n dx (a > 0, a

[3] cos 5 xdx

[4] sin 2 x cos 3 xdx

[5]

= 1, mn

= 0)

dx

sin x

Calculate the following indeﬁnite integrals.

[1] ln(x 2 1)dx

[2]

[3] (ax 2 + bx + c)e mx dx (abcm

[4] tan x +

1

+1 dx

e

x

= 0)

tan x 2 dx

1

[5] 1 sin xdx

[1] Tokyo Univ. of Science:

[2] Saitama Univ.:

ln x

(x+1) 2 dx

5

3 sin x+4 cos x dx

[3] Yokohama City Univ.:

1

3

1

x 2 +1 dx

π

2

[4] Daido Institute of Technology:

0

sin x+cos x dx

sin

3

x

3π

[5] Gunma Univ.:

0

4 {(1 + x) sin x + (1 x) cos x}dx

Evaluate

Evaluate

1

0

(1 x 2 ) n dx (n = 0, 1, 2, · · · )

aπ 0 a

lim

2

0

(cos x) ln(cos x) dx

Find the minimum value of

0

π (x y) 2 (sin x)| cos x|dx.

0 a < π

2

Let |a| < π 2 .

Evaluate

Evaluate

e

π

0

2

e e e e

dx

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

dx

e e

x ln x · ln(ln x) · ln{ln(ln x)}

Let f (x) = t sin x + (1 t) cos x (0 t 1).

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

Evaluate

P(t) = π e x f (x)dx π e x f (x)dx

0

2

0

2

π

0

4

x cos 5x

cos x

dx

Let a be a real number. Evaluate

3π+a

π+a

|x a π| sin x dx

2

A sequence {a n } is deﬁned by a n =

0

1 x 3 (1 x) n dx (n = 1, 2, 3. · · · )

Find the constant number c such that n=1 (n + c)(a n a n+1 ) =

1

3

Evaluate

Evaluate

n→∞ π

lim

0

x 2 | sin nx|dx

1

0

e x+e x +e e x +e e e x

dx

Evaluate

0

ln 2

dx

cos 2 hx · 1 2a tanh x + a 2

(a > 0)

Let p be a constant number such that 0 < p < 1. Evaluate

2004

k=0

p k (1 p) 2004k

1

0

x k (1 x) 2004k dx

1

Determine the value of a, b for which

0

( 1 x ax b) 2 dx is minimized.

A sequence of polynomial f n (x) (n = 0, 1, 2, · · · ) satisﬁes f 0 (x) = 2, f 1 (x) = x,

f n (x) = xf n1 (x) f n2 (x), (n = 2, 3, 4, ··· )

Let x n (n 2) be the maximum real root of the equation f n (x) = 0 (|x| 2)

Evaluate

Evaluate

n→∞ n 2

x

2

n

1

lim

f n (x)dx

π

2

2π

3

sin x 1 cos x

dx

Let a be a constant number such that 0 < a < 1 and V (a) be the volume formed by the revolution of the ﬁgure

which is enclosed by the curve y = ln(x a), the x-axis and two lines x = 1, x = 3 about the x-axis.

If a varies in the range of 0 < a < 1, ﬁnd the minimum value of V (a).

Find the minimum value of the following function f (x) deﬁned at 0 < x < π

2 .

f(x) = x

0

cos θ +

x

π

2

sin θ

Evaluate

Evaluate

Let 0 < t < π 2 .

Evaluate

Evaluate

Evaluate

a

0

1

0

x 2005 e x 2 dx

2ax x 2 dx (a > 0)

lim

tπ

2

t

0

tan θ cos θ ln(cos θ)

π

0

2

cos 2004 x cos 2004x dx

π

0

2

sin 2005x

sin x

dx

Find the function f (x) which satisﬁes the following integral equation.

f(x) = x t(sin t cos t)dt + π e t f (t)dt

0

0

2

Find the minimum value of

0

1 |tx|

t+1

dt

Find the condition of a, b for which the function f (x) (0 x 2π) satisfying the following equality can be determined uniquely,then determine f (x), assuming that f (x) is a continuous function at 0 x 2π.

f(x) =

2π 2π

a

0

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

2π 2π

b

0

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

48
Evaluate
n
sin 2 nx
1
lim
dx −
n→∞ π
sin x
k
0
k=1
x
1
49
For x ≥ 0, Prove that
(t − t 2 ) sin 2002 t <
0
2004·2005
50
Let a, b be real numbers such that a < b. Evaluate
b
ln |1 + (x − a)(b − x)|dx
a
lim
b→a
(b − a) 3
.
c
51
A
function f (x) satisﬁes f (x) = f x for some real number c(> 1) and all positive number
x.
c
f(x)
c
f(x)
If
dx = 3, evaluate
dx
1
x
1
x
52
Evaluate
n
1
lim
n→∞
n + k √ −1
k=1
53
Find the maximum value of the following integral.
e −x sin tx dx
0
54
evaluate
0
−1 1 + x
1 −
x dx
55
Evaluate
lim
(1 + x) −n−1 e x 2 dx
(n = 1, 2, · · · )
n→∞ n 1
0

Evaluate

[x] is the greatest integer x.

lim

n→∞

n

k=1

[ 2n 2 k 2 ]

n

2

Find the value of n N satisfying the following inequality.

Let f (x) =

Prove the following equation.

e

x

e x +1

π

0

x 2 sin nx dx

99π 2

<

100n

Evaluate

a

b

f (x)dx +

f(b)

f(a)

f 1 (x)dx = bf (b) af (a)

Let a n =

Evaluate

π

2

0

π

π

(cos 2x)(cos 2 2 x) · · · (cos 2 2006 x)dx

sin 2t (1 sin t) n1 dt (n = 1, 2, · · · )

2

Evaluate

For a > 1, let f (a) = f (a) at a > 1.

Evaluate

2

0

1

1

n=1

(n + 1)(a n a n+1 )

2004

k=0

1

1

1 x 2 k + 1 x

dx

1

|ax n 1|dx + 2 (n = 1, 2, · · · ) and let b n be the minimum value of

m b m · b m+1 · ······ b 2m (m = 1, 2, 3, ··· )

lim

For a positive number x, let f (x) = lim n n

Evaluate

k=1

cos 2k+1

2n

x cos 2k1

2n

lim

x→∞

f(x)

x

x

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

1

0

{f(t)} 2 1 t 2 dt

Let a > 0. Find the minimum value of

1 1 x

a

a

1 + x dx

π

2

Find the minimum value of

0

| cos x a| sin x dx

Evaluate

2005 1002

0

1002 2 x 2 + 1003 2 x 2 +

dx

1003

1002

1003 2 x 2 dx

1

0

1 x 2 dx

Find the minimum value of

Let f 1 (x) = x, f n (x) = x +

Find lim n f n (x)

e

ln x a dx (0 a e)

x

1

1 14 0

π

xf n1 (t) cos 3 t dt (n 2).

π

2

Find the number of root for

0

e x cos(x + a) dx = 0 at 0 a < 2π

Find the minimum value of 1 |t x| dt

1

Let f (x) be a continuous function satisfying f (x) = 1+k

Find the value of k for which

π

2

π

2

0

f (x)dx is maximized.

π

f (t) sin(xt)dt (k : constant number)

Find the minimum value of

π (a sin x + b sin 2x + c sin 3x x) 2 dx

0

74
p, q satisﬁes px + q ≥ ln x at a ≤ x ≤ b (0 < a < b).
Find the value of p, q for which the following deﬁnite integral is minimized and then the
minimum value.
b
(px + q − ln x)dx
a
75
A function f (θ) satisﬁes the following conditions (a), (b).
(a)
f(θ) ≥ 0
π
(b)
f (θ) sin θdθ = 1
0
Prove the following inequality.
π
f (θ) sin nθ dθ ≤ n (n = 1, 2, · · · )
0
76
The function f n (x) (n = 1, 2, · · · ) is deﬁned as follows.
f 1 (x) = x,
f n+1 (x) = 2x n+1 − x n + 1 2 1 f n (t) dt
(n = 1, 2, · · · )
0
Evaluate
1
lim
n→∞ f n 1 + 2n
77
Find the area of the part enclosed by the following curve.
x 2 + 2axy + y 2 = 1 (−1 < a < 1)
78
Let α, β be the distinct positive roots of the equation of 2x = tan x. Evaluate
1
sin αx sin βx dx
0
79
Find the area of the domain expressed by the following system inequalities.
1 1
x ≥ 0,
y ≥ 0,
x p + y p ≤ 1 (p = 1, 2, ··· )

Let S be the domain surrounded by the two curves C 1 : y = ax 2 , C 2 : y = ax 2 + 2abx for constant positive numbers a, b. Let V x be the volume of the solid formed by the revolution of S about the axis of x, V y be the volume of the solid formed by the revolution of S

about the axis of y. Find the ratio of V x

V y .

Prove the following inequality.

12 (π 6 + 2 3)

1

π

6

π

4

ln(1 + cos 2x)dx 1 4 (2 3)

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

Evaluate

Evaluate

b a

1

a

b

ln

b

x 2

n=1

2(n+1)π

2

x sin x + cos x

x

2

 dx < 2 dx (n = 1, 2, · · · )

Evaluate

n→∞ n π

lim

0

e nx sin 2 nx dx

n→∞

0

lim

π

2

[n sin x]

n

where [x] is the integer equal to x or less than x.

Prove

π

cos x

x

dx 2 <

dx

1

π 2

Find the minimum value of a (0 < a < 1) for which the following deﬁnite integral is minimized.

π

0

| sin x ax| dx

A function f (x) satisﬁes

f(x) = f (x) (4x 2)f (x)

f (0) = a,

f (1) = b

1

Evaluate

0

f (x)(x 2 x) dx.

For f (x) = x 4 + |x|,

Find the value of I 1

I 2 .

π

let I 1 =

0

f (cos x) dx,

π

2

I 2 =

0

f (sin x) dx.

Find lim n 3n

2n C n

C

n

1

n

where i C j is a binominal coeﬃcient which means i·(i1)···(ij+1)

j·(j1)···2·1

.

Prove the following inequality.

n=0 1

0

x 4011 (1 x 2006 )

2006 dx < 2006

2005

n1
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