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Today's Calculation Of Integral
2011 Today's Calculation Of Integral
2011 Today's Calculation Of Integral
Part of
Today's Calculation Of Integral
Subcontests
(95)
769
1
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Today's calculation of Integral 769
In
x
y
z
xyz
x
yz
space, find the volume of the solid expressed by
x
2
+
y
2
≤
z
≤
3
y
+
1.
x^2+y^2\leq z\le \sqrt{3}y+1.
x
2
+
y
2
≤
z
≤
3
y
+
1.
768
1
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Today's calculation of Integral 768
Let
r
r
r
be a real such that
0
<
r
≤
1
0<r\leq 1
0
<
r
≤
1
. Denote by
V
(
r
)
V(r)
V
(
r
)
the volume of the solid formed by all points of
(
x
,
y
,
z
)
(x,\ y,\ z)
(
x
,
y
,
z
)
satisfying
x
2
+
y
2
+
z
2
≤
1
,
x
2
+
y
2
≤
r
2
x^2+y^2+z^2\leq 1,\ x^2+y^2\leq r^2
x
2
+
y
2
+
z
2
≤
1
,
x
2
+
y
2
≤
r
2
in
x
y
z
xyz
x
yz
-space.(1) Find
V
(
r
)
V(r)
V
(
r
)
.(2) Find
lim
r
→
1
−
0
V
(
1
)
−
V
(
r
)
(
1
−
r
)
3
2
.
\lim_{r\rightarrow 1-0} \frac{V(1)-V(r)}{(1-r)^{\frac 32}}.
lim
r
→
1
−
0
(
1
−
r
)
2
3
V
(
1
)
−
V
(
r
)
.
(3) Find
lim
r
→
+
0
V
(
r
)
r
2
.
\lim_{r\rightarrow +0} \frac{V(r)}{r^2}.
lim
r
→
+
0
r
2
V
(
r
)
.
767
1
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Today's calculation of Integral 767
For
0
≤
t
≤
1
0\leq t\leq 1
0
≤
t
≤
1
, define
f
(
t
)
=
∫
0
2
π
∣
sin
x
−
t
∣
d
x
.
f(t)=\int_0^{2\pi} |\sin x-t|dx.
f
(
t
)
=
∫
0
2
π
∣
sin
x
−
t
∣
d
x
.
Evaluate
∫
0
1
f
(
t
)
d
t
.
\int_0^1 f(t)dt.
∫
0
1
f
(
t
)
d
t
.
766
1
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Today's calculation of Integral 766
Let
f
(
x
)
f(x)
f
(
x
)
be a continuous function defined on
0
≤
x
≤
π
0\leq x\leq \pi
0
≤
x
≤
π
and satisfies
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
and
{
∫
0
π
(
sin
x
+
cos
x
)
f
(
x
)
d
x
}
2
=
π
∫
0
π
{
f
(
x
)
}
2
d
x
.
\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.
{
∫
0
π
(
sin
x
+
cos
x
)
f
(
x
)
d
x
}
2
=
π
∫
0
π
{
f
(
x
)
}
2
d
x
.
Evaluate
∫
0
π
{
f
(
x
)
}
3
d
x
.
\int_0^{\pi} \{f(x)\}^3dx.
∫
0
π
{
f
(
x
)
}
3
d
x
.
765
1
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Today's calculation of Integral 765
Define two functions
g
(
x
)
,
f
(
x
)
(
x
≥
0
)
g(x),\ f(x)\ (x\geq 0)
g
(
x
)
,
f
(
x
)
(
x
≥
0
)
by
g
(
x
)
=
∫
0
x
e
−
t
2
d
t
,
f
(
x
)
=
∫
0
1
e
−
(
1
+
s
2
)
x
1
+
s
2
d
s
.
g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.
g
(
x
)
=
∫
0
x
e
−
t
2
d
t
,
f
(
x
)
=
∫
0
1
1
+
s
2
e
−
(
1
+
s
2
)
x
d
s
.
Now we know that
f
′
(
x
)
=
−
∫
0
1
e
−
(
1
+
s
2
)
x
d
s
.
f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.
f
′
(
x
)
=
−
∫
0
1
e
−
(
1
+
s
2
)
x
d
s
.
(1) Find
f
(
0
)
.
f(0).
f
(
0
)
.
(2) Show that
f
(
x
)
≤
π
4
e
−
x
(
x
≥
0
)
.
f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).
f
(
x
)
≤
4
π
e
−
x
(
x
≥
0
)
.
(3) Let
h
(
x
)
=
{
g
(
x
)
}
2
h(x)=\{g(\sqrt{x})\}^2
h
(
x
)
=
{
g
(
x
)
}
2
. Show that
f
′
(
x
)
=
−
h
′
(
x
)
.
f'(x)=-h'(x).
f
′
(
x
)
=
−
h
′
(
x
)
.
(4) Find
lim
x
→
+
∞
g
(
x
)
\lim_{x\rightarrow +\infty} g(x)
lim
x
→
+
∞
g
(
x
)
Please solve the problem without using Double Integral or Jacobian for those Japanese High School Students who don't study them.
763
1
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Today's calculation of Integral 763
Evaluate
∫
1
4
x
−
2
(
x
2
+
4
)
x
d
x
.
\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}dx.
∫
1
4
(
x
2
+
4
)
x
x
−
2
d
x
.
762
1
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Today's calculation of Integral 762
Define a function
f
n
(
x
)
(
n
=
0
,
1
,
2
,
⋯
)
f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)
f
n
(
x
)
(
n
=
0
,
1
,
2
,
⋯
)
by
f
0
(
x
)
=
sin
x
,
f
n
+
1
(
x
)
=
∫
0
π
2
f
n
′
(
t
)
sin
(
x
+
t
)
d
t
.
f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.
f
0
(
x
)
=
sin
x
,
f
n
+
1
(
x
)
=
∫
0
2
π
f
n
′
(
t
)
sin
(
x
+
t
)
d
t
.
(1) Let
f
n
(
x
)
=
a
n
sin
x
+
b
n
cos
x
.
f_n(x)=a_n\sin x+b_n\cos x.
f
n
(
x
)
=
a
n
sin
x
+
b
n
cos
x
.
Express
a
n
+
1
,
b
n
+
1
a_{n+1},\ b_{n+1}
a
n
+
1
,
b
n
+
1
in terms of
a
n
,
b
n
.
a_n,\ b_n.
a
n
,
b
n
.
(2) Find
∑
n
=
0
∞
f
n
(
π
4
)
.
\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).
∑
n
=
0
∞
f
n
(
4
π
)
.
761
1
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Today's calculation of Integral 761
Find
lim
n
→
∞
1
n
(
4
n
)
!
(
3
n
)
!
n
.
\lim_{n\to\infty} \frac{1}{n}\sqrt[n]{\frac{(4n)!}{(3n)!}}.
lim
n
→
∞
n
1
n
(
3
n
)!
(
4
n
)!
.
760
1
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Today's calculation of Integral 760
Prove that there exists a positive integer
n
n
n
such that
∫
0
1
x
sin
(
x
2
−
x
+
1
)
d
x
≥
n
n
+
1
sin
n
+
2
n
+
3
.
\int_0^1 x\sin\ (x^2-x+1)dx\geq \frac {n}{n+1}\sin \frac{n+2}{n+3}.
∫
0
1
x
sin
(
x
2
−
x
+
1
)
d
x
≥
n
+
1
n
sin
n
+
3
n
+
2
.
759
1
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Today's calculation of Integral 759
Given a regular tetrahedron
P
Q
R
S
PQRS
PQRS
with side length
d
d
d
. Find the volume of the solid generated by a rotation around the line passing through
P
P
P
and the midpoint
M
M
M
of
Q
R
QR
QR
.
758
1
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Today's calculation of Integral 758
Find the slope of a line passing through the point
(
0
,
1
)
(0,\ 1)
(
0
,
1
)
with which the area of the part bounded by the line and the parabola
y
=
x
2
y=x^2
y
=
x
2
is
5
5
6
.
\frac{5\sqrt{5}}{6}.
6
5
5
.
757
1
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Today's calculation of Integral 757
Evaluate
∫
0
1
(
x
2
+
x
+
1
)
3
{
ln
(
x
2
+
x
+
1
)
+
2
}
(
x
2
+
x
+
1
)
3
(
2
x
+
1
)
e
x
2
+
x
+
1
d
x
.
\int_0^1 \frac{(x^2+x+1)^3\{\ln (x^2+x+1)+2\}}{(x^2+x+1)^3}(2x+1)e^{x^2+x+1}dx.
∫
0
1
(
x
2
+
x
+
1
)
3
(
x
2
+
x
+
1
)
3
{
ln
(
x
2
+
x
+
1
)
+
2
}
(
2
x
+
1
)
e
x
2
+
x
+
1
d
x
.
756
1
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Today's calculation of Integral 756
Let
a
a
a
be real number. A circle
C
C
C
touches the line
y
=
−
x
y=-x
y
=
−
x
at the point
(
a
,
−
a
)
(a, -a)
(
a
,
−
a
)
and passes through the point
(
0
,
1
)
.
(0,\ 1).
(
0
,
1
)
.
Denote by
P
P
P
the center of
C
C
C
. When
a
a
a
moves, find the area of the figure enclosed by the locus of
P
P
P
and the line
y
=
1
y=1
y
=
1
.
755
1
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Today's calculation of Integral 755
Given mobile points
P
(
0
,
sin
θ
)
,
Q
(
8
cos
θ
,
0
)
(
0
≤
θ
≤
π
2
)
P(0,\ \sin \theta),\ Q(8\cos \theta,\ 0)\ \left(0\leq \theta \leq \frac{\pi}{2}\right)
P
(
0
,
sin
θ
)
,
Q
(
8
cos
θ
,
0
)
(
0
≤
θ
≤
2
π
)
on the
x
x
x
-
y
y
y
plane. Denote by
D
D
D
the part in which line segment
P
Q
PQ
PQ
sweeps. Find the volume
V
V
V
generated by a rotation of
D
D
D
around the
x
x
x
-axis.
754
1
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Today's calculation of Integral 754
Let
S
n
S_n
S
n
be the area of the figure enclosed by a curve
y
=
x
2
(
1
−
x
)
n
(
0
≤
x
≤
1
)
y=x^2(1-x)^n\ (0\leq x\leq 1)
y
=
x
2
(
1
−
x
)
n
(
0
≤
x
≤
1
)
and the
x
x
x
-axis. Find
lim
n
→
∞
∑
k
=
1
n
S
k
.
\lim_{n\to\infty} \sum_{k=1}^n S_k.
lim
n
→
∞
∑
k
=
1
n
S
k
.
753
1
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Today's calculation of Integral 753
Find
lim
n
→
∞
∑
k
=
1
2
n
n
2
n
2
+
3
n
k
+
k
2
.
\lim_{n\to\infty} \sum_{k=1}^{2n} \frac{n}{2n^2+3nk+k^2}.
lim
n
→
∞
∑
k
=
1
2
n
2
n
2
+
3
nk
+
k
2
n
.
752
1
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Today's calculation of Integral 752
Find
f
n
(
x
)
f_n(x)
f
n
(
x
)
such that
f
1
(
x
)
=
x
,
f
n
(
x
)
=
∫
0
x
t
f
n
−
1
(
x
−
t
)
d
t
(
n
=
2
,
3
,
⋯
)
.
f_1(x)=x,\ f_n(x)=\int_0^x tf_{n-1}(x-t)dt\ (n=2,\ 3,\ \cdots).
f
1
(
x
)
=
x
,
f
n
(
x
)
=
∫
0
x
t
f
n
−
1
(
x
−
t
)
d
t
(
n
=
2
,
3
,
⋯
)
.
751
1
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Today's calculation of Integral 751
Find
lim
n
→
∞
(
1
n
∫
0
n
(
sin
2
π
x
)
ln
(
x
+
n
)
d
x
−
1
2
ln
n
)
.
\lim_{n\to\infty}\left(\frac{1}{n}\int_0^n (\sin ^ 2 \pi x)\ln (x+n)dx-\frac 12\ln n\right).
lim
n
→
∞
(
n
1
∫
0
n
(
sin
2
π
x
)
ln
(
x
+
n
)
d
x
−
2
1
ln
n
)
.
750
1
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Today's calculation of Integral 750
Let
a
n
(
n
≥
1
)
a_n\ (n\geq 1)
a
n
(
n
≥
1
)
be the value for which
∫
x
2
x
e
−
t
n
d
t
(
x
≥
0
)
\int_x^{2x} e^{-t^n}dt\ (x\geq 0)
∫
x
2
x
e
−
t
n
d
t
(
x
≥
0
)
is maximal. Find
lim
n
→
∞
ln
a
n
.
\lim_{n\to\infty} \ln a_n.
lim
n
→
∞
ln
a
n
.
749
1
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Today's calculation of Integral 749
Let
m
m
m
be a positive integer. A tangent line at the point
P
P
P
on the parabola
C
1
:
y
=
x
2
+
m
2
C_1 : y=x^2+m^2
C
1
:
y
=
x
2
+
m
2
intersects with the parabola
C
2
:
y
=
x
2
C_2 : y=x^2
C
2
:
y
=
x
2
at the points
A
,
B
A,\ B
A
,
B
. For the point
Q
Q
Q
between
A
A
A
and
B
B
B
on
C
2
C_2
C
2
, denote by
S
S
S
the sum of the areas of the region bounded by the line
A
Q
AQ
A
Q
,
C
2
C_2
C
2
and the region bounded by the line
Q
B
QB
QB
,
C
2
C_2
C
2
. When
Q
Q
Q
move between
A
A
A
and
B
B
B
on
C
2
C_2
C
2
, prove that the minimum value of
S
S
S
doesn't depend on how we would take
P
P
P
, then find the value in terms of
m
m
m
.
748
1
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Today's calculation of Integral 748
Evaluate the following integrals.(1)
∫
0
π
cos
m
x
cos
n
x
d
x
(
m
,
n
=
1
,
2
,
⋯
)
.
\int_0^{\pi} \cos mx\cos nx\ dx\ (m,\ n=1,\ 2,\ \cdots).
∫
0
π
cos
m
x
cos
n
x
d
x
(
m
,
n
=
1
,
2
,
⋯
)
.
(2)
∫
1
3
(
x
−
1
x
)
(
ln
x
)
2
d
x
.
\int_1^3 \left(x-\frac{1}{x}\right)(\ln x)^2dx.
∫
1
3
(
x
−
x
1
)
(
ln
x
)
2
d
x
.
747
1
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Today's calculation of Integral 747
Prove that
∫
0
4
(
1
−
cos
x
2
)
e
x
d
x
≤
−
2
e
2
+
30.
\int_0^4 \left(1-\cos \frac{x}{2}\right)e^{\sqrt{x}}dx\leq -2e^2+30.
∫
0
4
(
1
−
cos
2
x
)
e
x
d
x
≤
−
2
e
2
+
30.
746
1
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Today's calculation of Integral 746
Prove the following inequality.
n
n
e
−
n
+
1
≤
n
!
≤
1
4
(
n
+
1
)
n
+
1
e
−
n
+
1
.
n^ne^{-n+1}\leq n!\leq \frac 14(n+1)^{n+1}e^{-n+1}.
n
n
e
−
n
+
1
≤
n
!
≤
4
1
(
n
+
1
)
n
+
1
e
−
n
+
1
.
745
1
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Today's calculation of Integral 745
When real numbers
a
,
b
a,\ b
a
,
b
move satisfying
∫
0
π
(
a
cos
x
+
b
sin
x
)
2
d
x
=
1
\int_0^{\pi} (a\cos x+b\sin x)^2dx=1
∫
0
π
(
a
cos
x
+
b
sin
x
)
2
d
x
=
1
, find the maximum value of
∫
0
π
(
e
x
−
a
cos
x
−
b
sin
x
)
2
d
x
.
\int_0^{\pi} (e^x-a\cos x-b\sin x)^2dx.
∫
0
π
(
e
x
−
a
cos
x
−
b
sin
x
)
2
d
x
.
744
1
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Today's calculation of Integral 744
Let
a
,
b
a,\ b
a
,
b
be real numbers. If
∫
0
3
(
a
x
−
b
)
2
d
x
≤
3
\int_0^3 (ax-b)^2dx\leq 3
∫
0
3
(
a
x
−
b
)
2
d
x
≤
3
holds, then find the values of
a
,
b
a,\ b
a
,
b
such that
∫
0
3
(
x
−
3
)
(
a
x
−
b
)
d
x
\int_0^3 (x-3)(ax-b)dx
∫
0
3
(
x
−
3
)
(
a
x
−
b
)
d
x
is minimized.
743
1
Hide problems
Today's calculation of Integral 743
Evaluate
∫
0
π
2
ln
(
1
+
sin
θ
3
)
cos
θ
d
θ
.
\int_0^{\frac{\pi}{2}} \ln (1+\sqrt[3]{\sin \theta})\cos \theta\ d\theta.
∫
0
2
π
ln
(
1
+
3
sin
θ
)
cos
θ
d
θ
.
742
1
Hide problems
Today's calculation of Integral 742
Evaluate
∫
0
1
1
−
x
2
(
1
+
x
2
)
1
+
x
4
d
x
\int_0^1 \frac{1-x^2}{(1+x^2)\sqrt{1+x^4}}\ dx
∫
0
1
(
1
+
x
2
)
1
+
x
4
1
−
x
2
d
x
741
1
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Today's calculation of Integral 741
Evaluate
∫
0
1
(
x
−
1
)
2
(
cos
x
+
1
)
−
(
2
x
−
1
)
sin
x
(
x
−
1
+
sin
x
)
2
d
x
\int_0^1 \frac{(x-1)^2(\cos x+1)-(2x-1)\sin x}{(x-1+\sqrt{\sin x})^2}\ dx
∫
0
1
(
x
−
1
+
sin
x
)
2
(
x
−
1
)
2
(
cos
x
+
1
)
−
(
2
x
−
1
)
sin
x
d
x
740
1
Hide problems
Today's calculation of Integral 740
Let
r
r
r
be a positive constant. If 2 curves
C
1
:
y
=
2
x
2
x
2
+
1
,
C
2
:
y
=
r
2
−
x
2
C_1: y=\frac{2x^2}{x^2+1},\ C_2: y=\sqrt{r^2-x^2}
C
1
:
y
=
x
2
+
1
2
x
2
,
C
2
:
y
=
r
2
−
x
2
have each tangent line at their point of intersection and at which their tangent lines are perpendicular each other, then find the area of the figure bounded by
C
1
,
C
2
C_1,\ C_2
C
1
,
C
2
.
739
1
Hide problems
Today's calculation of Integral 739
Find the function
f
(
x
)
f(x)
f
(
x
)
such that :
f
(
x
)
=
cos
x
+
∫
0
2
π
f
(
y
)
sin
(
x
−
y
)
d
y
f(x)=\cos x+\int_0^{2\pi} f(y)\sin (x-y)\ dy
f
(
x
)
=
cos
x
+
∫
0
2
π
f
(
y
)
sin
(
x
−
y
)
d
y
738
1
Hide problems
Today's calculation of Integral 738
Answer the following questions:(1) Find the value of
a
a
a
for which
S
=
∫
−
π
π
(
x
−
a
sin
3
x
)
2
d
x
S=\int_{-\pi}^{\pi} (x-a\sin 3x)^2dx
S
=
∫
−
π
π
(
x
−
a
sin
3
x
)
2
d
x
is minimized, then find the minimum value.(2) Find the vlues of
p
,
q
p,\ q
p
,
q
for which
T
=
∫
−
π
π
(
sin
3
x
−
p
x
−
q
x
2
)
2
d
x
T=\int_{-\pi}^{\pi} (\sin 3x-px-qx^2)^2dx
T
=
∫
−
π
π
(
sin
3
x
−
p
x
−
q
x
2
)
2
d
x
is minimized, then find the minimum value.
737
1
Hide problems
Today's calculation of Integral 737
Let
a
,
b
a,\ b
a
,
b
real numbers such that
a
>
1
,
b
>
1.
a>1,\ b>1.
a
>
1
,
b
>
1.
Prove the following inequality.
∫
−
1
1
(
1
+
b
∣
x
∣
1
+
a
x
+
1
+
a
∣
x
∣
1
+
b
x
)
d
x
<
a
+
b
+
2
\int_{-1}^1 \left(\frac{1+b^{|x|}}{1+a^{x}}+\frac{1+a^{|x|}}{1+b^{x}}\right)\ dx<a+b+2
∫
−
1
1
(
1
+
a
x
1
+
b
∣
x
∣
+
1
+
b
x
1
+
a
∣
x
∣
)
d
x
<
a
+
b
+
2
736
1
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Today's calculation of Integral 736
Evaluate
∫
0
1
(
e
x
+
1
)
{
e
x
+
1
+
(
1
+
x
+
e
x
)
ln
(
1
+
x
+
e
x
)
}
1
+
x
+
e
x
d
x
\int_0^1 \frac{(e^x+1)\{e^x+1+(1+x+e^x)\ln (1+x+e^x)\}}{1+x+e^x}\ dx
∫
0
1
1
+
x
+
e
x
(
e
x
+
1
)
{
e
x
+
1
+
(
1
+
x
+
e
x
)
ln
(
1
+
x
+
e
x
)}
d
x
735
1
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Today's calculation of Integral 735
Evaluate the following definite integrals:(a)
∫
0
π
2
x
tan
(
x
2
)
d
x
\int_0^{\frac{\sqrt{\pi}}{2}} x\tan (x^2)\ dx
∫
0
2
π
x
tan
(
x
2
)
d
x
(b)
∫
0
1
3
x
e
3
x
d
x
\int_0^{\frac 13} xe^{3x}\ dx
∫
0
3
1
x
e
3
x
d
x
(c)
∫
e
e
e
1
x
ln
x
d
x
\int_e^{e^e} \frac{1}{x\ln x}\ dx
∫
e
e
e
x
l
n
x
1
d
x
(d)
∫
2
3
x
2
+
1
x
(
x
+
1
)
d
x
\int_2^3 \frac{x^2+1}{x(x+1)}\ dx
∫
2
3
x
(
x
+
1
)
x
2
+
1
d
x
734
1
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Today's calculation of Integral 734
Find the extremum of
f
(
t
)
=
∫
1
t
ln
x
x
+
t
d
x
(
t
>
0
)
f(t)=\int_1^t \frac{\ln x}{x+t}dx\ (t>0)
f
(
t
)
=
∫
1
t
x
+
t
l
n
x
d
x
(
t
>
0
)
.
733
1
Hide problems
Today's calculation of Integral 733
Find
lim
n
→
∞
∫
0
1
x
2
e
−
(
x
n
)
2
d
x
.
\lim_{n\to\infty} \int_0^1 x^2e^{-\left(\frac{x}{n}\right)^2}dx.
lim
n
→
∞
∫
0
1
x
2
e
−
(
n
x
)
2
d
x
.
732
1
Hide problems
Today's calculation of Integral 732
Let
a
a
a
be parameter such that
0
<
a
<
2
π
0<a<2\pi
0
<
a
<
2
π
. For
0
<
x
<
2
π
0<x<2\pi
0
<
x
<
2
π
, find the extremum of
F
(
x
)
=
∫
x
x
+
a
1
−
cos
θ
d
θ
F(x)=\int_{x}^{x+a} \sqrt{1-\cos \theta}\ d\theta
F
(
x
)
=
∫
x
x
+
a
1
−
cos
θ
d
θ
.
731
1
Hide problems
Today's calculation of Integral 731
Let
C
C
C
be the point of intersection of the tangent lines
l
,
m
l,\ m
l
,
m
at A(a,\ a^2),\ B(b,\ b^2)\ (a
y
=
x
2
y=x^2
y
=
x
2
respectively. When
C
C
C
moves on the parabola
y
=
1
2
x
2
−
x
−
2
y=\frac 12x^2-x-2
y
=
2
1
x
2
−
x
−
2
, find the minimum area bounded by 2 lines
l
,
m
l,\ m
l
,
m
and the parabola
y
=
x
2
y=x^2
y
=
x
2
.
730
1
Hide problems
Today's calculation of Integral 730
Let
a
n
a_n
a
n
be the local maximum of
f
n
(
x
)
=
x
n
e
−
x
+
n
π
n
!
(
n
=
1
,
2
,
⋯
)
f_n(x)=\frac{x^ne^{-x+n\pi}}{n!}\ (n=1,\ 2,\ \cdots)
f
n
(
x
)
=
n
!
x
n
e
−
x
+
nπ
(
n
=
1
,
2
,
⋯
)
for
x
>
0
x>0
x
>
0
.Find
lim
n
→
∞
ln
(
a
2
n
a
n
)
1
n
\lim_{n\to\infty} \ln \left(\frac{a_{2n}}{a_n}\right)^{\frac{1}{n}}
lim
n
→
∞
ln
(
a
n
a
2
n
)
n
1
.
729
1
Hide problems
Today's calculation of Integral 729
Evaluate
∫
1
e
ln
x
−
1
x
2
−
(
ln
x
)
2
d
x
.
\int_1^e \frac{\ln x-1}{x^2-(\ln x)^2}dx.
∫
1
e
x
2
−
(
l
n
x
)
2
l
n
x
−
1
d
x
.
728
1
Hide problems
Today's calculation of Integral 728
Evaluate
∫
π
12
π
6
sin
x
−
cos
x
−
x
(
sin
x
+
cos
x
)
+
1
x
2
−
x
(
sin
x
+
cos
x
)
+
sin
x
cos
x
d
x
.
\int_{\frac {\pi}{12}}^{\frac{\pi}{6}} \frac{\sin x-\cos x-x(\sin x+\cos x)+1}{x^2-x(\sin x+\cos x)+\sin x\cos x}\ dx.
∫
12
π
6
π
x
2
−
x
(
sin
x
+
cos
x
)
+
sin
x
cos
x
sin
x
−
cos
x
−
x
(
sin
x
+
cos
x
)
+
1
d
x
.
727
1
Hide problems
Today's calculation of Integral 727
For positive constant
a
a
a
, let
C
:
y
=
a
2
(
e
x
a
+
e
−
x
a
)
C: y=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})
C
:
y
=
2
a
(
e
a
x
+
e
−
a
x
)
. Denote by
l
(
t
)
l(t)
l
(
t
)
the length of the part
a
≤
y
≤
t
a\leq y\leq t
a
≤
y
≤
t
for
C
C
C
and denote by
S
(
t
)
S(t)
S
(
t
)
the area of the part bounded by the line
y
=
t
(
a
<
t
)
y=t\ (a<t)
y
=
t
(
a
<
t
)
and
C
C
C
. Find
lim
t
→
∞
S
(
t
)
l
(
t
)
ln
t
.
\lim_{t\to\infty} \frac{S(t)}{l(t)\ln t}.
lim
t
→
∞
l
(
t
)
l
n
t
S
(
t
)
.
726
1
Hide problems
Today's calculation of Integral 726
Let
P
(
x
,
y
)
(
x
>
0
,
y
>
0
)
P(x,\ y)\ (x>0,\ y>0)
P
(
x
,
y
)
(
x
>
0
,
y
>
0
)
be a point on the curve
C
:
x
2
−
y
2
=
1
C: x^2-y^2=1
C
:
x
2
−
y
2
=
1
. If
x
=
e
u
+
e
−
u
2
(
u
≥
0
)
x=\frac{e^u+e^{-u}}{2}\ (u\geq 0)
x
=
2
e
u
+
e
−
u
(
u
≥
0
)
, then find the area bounded by the line
O
P
OP
OP
, the
x
x
x
axis and the curve
C
C
C
in terms of
u
u
u
.
725
1
Hide problems
Today's calculation of Integral 725
For
a
>
1
a>1
a
>
1
, evaluate
∫
1
a
a
1
x
(
ln
x
)
ln
(
x
2
+
1
)
d
x
.
\int_{\frac{1}{a}}^a \frac{1}{x}(\ln x)\ln\ (x^2+1)dx.
∫
a
1
a
x
1
(
ln
x
)
ln
(
x
2
+
1
)
d
x
.
724
1
Hide problems
Today's calculation of Integral 724
Find
lim
n
→
∞
{
(
1
+
n
)
1
n
(
1
+
n
2
)
2
n
(
1
+
n
3
)
3
n
⋯
⋯
2
}
1
n
\lim_{n\to\infty}\left\{\left(1+n\right)^{\frac{1}{n}}\left(1+\frac{n}{2}\right)^{\frac{2}{n}}\left(1+\frac{n}{3}\right)^{\frac{3}{n}}\cdots\cdots 2\right\}^{\frac{1}{n}}
lim
n
→
∞
{
(
1
+
n
)
n
1
(
1
+
2
n
)
n
2
(
1
+
3
n
)
n
3
⋯⋯
2
}
n
1
.
723
1
Hide problems
Today's calculation of Integral 723
Evaluate
∫
1
e
{
1
−
(
x
−
1
)
e
x
}
ln
x
(
1
+
e
x
)
2
d
x
.
\int_1^e \frac{\{1-(x-1)e^{x}\}\ln x}{(1+e^x)^2}dx.
∫
1
e
(
1
+
e
x
)
2
{
1
−
(
x
−
1
)
e
x
}
l
n
x
d
x
.
722
1
Hide problems
Today's calculation of Integral 722
Find the continuous function
f
(
x
)
f(x)
f
(
x
)
such that :
∫
0
x
f
(
t
)
(
∫
0
t
f
(
t
)
d
t
)
d
t
=
f
(
x
)
+
1
2
\int_0^x f(t)\left(\int_0^t f(t)dt\right)dt=f(x)+\frac 12
∫
0
x
f
(
t
)
(
∫
0
t
f
(
t
)
d
t
)
d
t
=
f
(
x
)
+
2
1
721
1
Hide problems
Today's calculation of Integral 721
For constant
a
a
a
, find the differentiable function
f
(
x
)
f(x)
f
(
x
)
satisfying
∫
0
x
(
e
−
x
−
a
e
−
t
)
f
(
t
)
d
t
=
0
\int_0^x (e^{-x}-ae^{-t})f(t)dt=0
∫
0
x
(
e
−
x
−
a
e
−
t
)
f
(
t
)
d
t
=
0
.
720
1
Hide problems
Today's calculation of Integral 720
Evaluate
∫
0
2
π
∣
x
2
−
π
2
−
sin
2
x
∣
d
x
\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx
∫
0
2
π
∣
x
2
−
π
2
−
sin
2
x
∣
d
x
.
719
1
Hide problems
Today's calculation of Integral 719
Compute
∫
0
x
sin
t
cos
t
sin
(
2
π
cos
t
)
d
t
\int_0^x \sin t\cos t\sin (2\pi\cos t)\ dt
∫
0
x
sin
t
cos
t
sin
(
2
π
cos
t
)
d
t
.
718
1
Hide problems
Today's calculation of Integral 718
Find
∑
n
=
1
∞
1
2
n
∫
−
1
1
(
1
−
x
)
2
(
1
+
x
)
n
d
x
(
n
≥
1
)
.
\sum_{n=1}^{\infty} \frac{1}{2^n}\int_{-1}^1 (1-x)^2(1+x)^n dx\ (n\geq 1).
∑
n
=
1
∞
2
n
1
∫
−
1
1
(
1
−
x
)
2
(
1
+
x
)
n
d
x
(
n
≥
1
)
.
717
1
Hide problems
Today's calculation of Integral 717
Let
a
n
a_n
a
n
be the area of the part enclosed by the curve
y
=
x
n
(
n
≥
1
)
y=x^n\ (n\geq 1)
y
=
x
n
(
n
≥
1
)
, the line
x
=
1
2
x=\frac 12
x
=
2
1
and the
x
x
x
axis. Prove that :
0
≤
ln
2
−
1
2
−
(
a
1
+
a
2
+
⋯
⋯
+
a
n
)
≤
1
2
n
+
1
0\leq \ln 2-\frac 12-(a_1+a_2+\cdots\cdots+a_n)\leq \frac {1}{2^{n+1}}
0
≤
ln
2
−
2
1
−
(
a
1
+
a
2
+
⋯⋯
+
a
n
)
≤
2
n
+
1
1
716
1
Hide problems
Today's calculation of Integral 716
Prove that :
∫
1
e
(
ln
x
)
n
d
x
=
(
−
1
)
n
−
1
n
!
+
e
∑
m
=
0
n
(
−
1
)
n
−
m
n
!
m
!
(
1
2
)
m
\int_1^{\sqrt{e}} (\ln x)^n\ dx=(-1)^{n-1}n!+\sqrt{e}\sum_{m=0}^{n} (-1)^{n-m}\frac{n!}{m!}\left(\frac 12\right)^{m}
∫
1
e
(
ln
x
)
n
d
x
=
(
−
1
)
n
−
1
n
!
+
e
m
=
0
∑
n
(
−
1
)
n
−
m
m
!
n
!
(
2
1
)
m
715
1
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Today's calculation of Integral 715
Find the differentiable function
f
(
x
)
f(x)
f
(
x
)
with
f
(
0
)
≠
0
f(0)\neq 0
f
(
0
)
=
0
satisfying
f
(
x
+
y
)
=
f
(
x
)
f
′
(
y
)
+
f
′
(
x
)
f
(
y
)
f(x+y)=f(x)f'(y)+f'(x)f(y)
f
(
x
+
y
)
=
f
(
x
)
f
′
(
y
)
+
f
′
(
x
)
f
(
y
)
for all real numbers
x
,
y
x,\ y
x
,
y
.
714
1
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Today's calculation of Integral 714
Find the area enclosed by the graph of
a
2
x
4
=
b
2
x
2
−
y
2
(
a
>
0
,
b
>
0
)
.
a^2x^4=b^2x^2-y^2\ (a>0,\ b>0).
a
2
x
4
=
b
2
x
2
−
y
2
(
a
>
0
,
b
>
0
)
.
713
1
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Today's calculation of Integral 713
If a positive sequence
{
a
n
}
n
≥
1
\{a_n\}_{n\geq 1}
{
a
n
}
n
≥
1
satisfies
∫
0
a
n
x
n
d
x
=
2
\int_0^{a_n} x^{n}\ dx=2
∫
0
a
n
x
n
d
x
=
2
, then find
lim
n
→
∞
a
n
.
\lim_{n\to\infty} a_n.
lim
n
→
∞
a
n
.
712
1
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Today's calculation of Integral 712
Evaluate
∫
π
4
π
3
{
1
tan
x
(
ln
sin
x
)
+
tan
x
ln
cos
x
}
d
x
.
\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \left\{\frac{1}{\tan x\ (\ln \sin x)}+\frac{\tan x}{\ln \cos x}\right\}\ dx.
∫
4
π
3
π
{
t
a
n
x
(
l
n
s
i
n
x
)
1
+
l
n
c
o
s
x
t
a
n
x
}
d
x
.
711
1
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Today's calculation of Integral 711
Evaluate
∫
e
e
2
4
(
ln
x
)
2
+
1
(
ln
x
)
3
2
d
x
.
\int_e^{e^2} \frac{4(\ln x)^2+1}{(\ln x)^{\frac 32}}\ dx.
∫
e
e
2
(
l
n
x
)
2
3
4
(
l
n
x
)
2
+
1
d
x
.
710
1
Hide problems
Today's calculation of Integral 710
Evaluate
∫
0
π
4
sin
θ
(
sin
θ
cos
θ
+
2
)
cos
4
θ
d
θ
\int_0^{\frac{\pi}{4}} \frac{\sin \theta (\sin \theta \cos \theta +2)}{\cos ^ 4 \theta}\ d\theta
∫
0
4
π
c
o
s
4
θ
s
i
n
θ
(
s
i
n
θ
c
o
s
θ
+
2
)
d
θ
.
706
1
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Today's calculation of Integral 706
In the
x
y
z
xyz
x
yz
space, consider a right circular cylinder with radius of base 2, altitude 4 such that
{
x
2
+
y
2
≤
4
e
m
s
p
;
0
≤
z
≤
4
e
m
s
p
;
\left\{ \begin{array}{ll} x^2+y^2\leq 4 &  \\ 0\leq z\leq 4 &  \end{array} \right.
{
x
2
+
y
2
≤
4
0
≤
z
≤
4
e
m
s
p
;
e
m
s
p
;
Let
V
V
V
be the solid formed by the points
(
x
,
y
,
z
)
(x,\ y,\ z)
(
x
,
y
,
z
)
in the circular cylinder satisfying
{
z
≤
(
x
−
2
)
2
e
m
s
p
;
z
≤
y
2
e
m
s
p
;
\left\{ \begin{array}{ll} z\leq (x-2)^2 &  \\ z\leq y^2 &  \end{array} \right.
{
z
≤
(
x
−
2
)
2
z
≤
y
2
e
m
s
p
;
e
m
s
p
;
Find the volume of the solid
V
V
V
.
705
1
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Today's calculation of Integral 705
The parametric equations of a curve are given by
x
=
2
(
1
+
cos
t
)
cos
t
,
y
=
2
(
1
+
cos
t
)
sin
t
(
0
≤
t
≤
2
π
)
x = 2(1+\cos t)\cos t,\ y =2(1+\cos t)\sin t\ (0\leq t\leq 2\pi)
x
=
2
(
1
+
cos
t
)
cos
t
,
y
=
2
(
1
+
cos
t
)
sin
t
(
0
≤
t
≤
2
π
)
.(1) Find the maximum and minimum values of
x
x
x
.(2) Find the volume of the solid enclosed by the figure of revolution about the
x
x
x
-axis.
704
1
Hide problems
Today's calculation of Integral 704
A function
f
n
(
x
)
(
n
=
0
,
1
,
2
,
3
,
⋯
)
f_n(x)\ (n=0,\ 1,\ 2,\ 3,\ \cdots)
f
n
(
x
)
(
n
=
0
,
1
,
2
,
3
,
⋯
)
satisfies the following conditions:(i)
f
0
(
x
)
=
e
2
x
+
1
f_0(x)=e^{2x}+1
f
0
(
x
)
=
e
2
x
+
1
.(ii)
f
n
(
x
)
=
∫
0
x
(
n
+
2
t
)
f
n
−
1
(
t
)
d
t
−
2
x
n
+
1
n
+
1
(
n
=
1
,
2
,
3
,
⋯
)
.
f_n(x)=\int_0^x (n+2t)f_{n-1}(t)dt-\frac{2x^{n+1}}{n+1}\ (n=1,\ 2,\ 3,\ \cdots).
f
n
(
x
)
=
∫
0
x
(
n
+
2
t
)
f
n
−
1
(
t
)
d
t
−
n
+
1
2
x
n
+
1
(
n
=
1
,
2
,
3
,
⋯
)
.
Find
∑
n
=
1
∞
f
n
′
(
1
2
)
.
\sum_{n=1}^{\infty} f_n'\left(\frac 12\right).
∑
n
=
1
∞
f
n
′
(
2
1
)
.
703
1
Hide problems
Today's calculation of Integral 703
Given a line segment
P
Q
PQ
PQ
with endpoints on the parabola
y
=
x
2
y=x^2
y
=
x
2
such that the area bounded by
P
Q
PQ
PQ
and the parabola always equal to
4
3
.
\frac 43.
3
4
.
Find the equation of the locus of the midpoint
M
M
M
.
702
1
Hide problems
Today's calculation of Integral 702
f
(
x
)
f(x)
f
(
x
)
is a continuous function defined in
x
>
0
x>0
x
>
0
. For all
a
,
b
(
a
>
0
,
b
>
0
)
a,\ b\ (a>0,\ b>0)
a
,
b
(
a
>
0
,
b
>
0
)
, if
∫
a
b
f
(
x
)
d
x
\int_a^b f(x)\ dx
∫
a
b
f
(
x
)
d
x
is determined by only
b
a
\frac{b}{a}
a
b
, then prove that
f
(
x
)
=
c
x
(
c
:
c
o
n
s
t
a
n
t
)
.
f(x)=\frac{c}{x}\ (c: constant).
f
(
x
)
=
x
c
(
c
:
co
n
s
t
an
t
)
.
701
1
Hide problems
Today's calculation of Integral 701
Evaluate
∫
π
4
π
2
(
1
+
cos
x
)
{
1
−
tan
2
x
2
tan
(
x
+
sin
x
)
tan
(
x
−
sin
x
)
}
tan
(
x
+
sin
x
)
d
x
\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{(1+\cos x)\{1-\tan ^ 2 \frac{x}{2}\tan (x+\sin x)\tan (x-\sin x)\}}{\tan (x+\sin x)}\ dx
∫
4
π
2
π
tan
(
x
+
sin
x
)
(
1
+
cos
x
)
{
1
−
tan
2
2
x
tan
(
x
+
sin
x
)
tan
(
x
−
sin
x
)}
d
x
700
1
Hide problems
Today's calculation of Integral 700
Evaluate
∫
0
π
x
2
cos
2
x
−
x
sin
x
−
cos
x
−
1
(
1
+
x
sin
x
)
2
d
x
\int_0^{\pi} \frac{x^2\cos ^ 2 x-x\sin x-\cos x-1}{(1+x\sin x)^2}dx
∫
0
π
(
1
+
x
sin
x
)
2
x
2
cos
2
x
−
x
sin
x
−
cos
x
−
1
d
x
699
1
Hide problems
Today's calculation of Integral 699
Find the volume of the part bounded by
z
=
x
+
y
,
z
=
x
2
+
y
2
z=x+y,\ z=x^2+y^2
z
=
x
+
y
,
z
=
x
2
+
y
2
in the
x
y
z
xyz
x
yz
space.
698
1
Hide problems
Today's calculation of Integral 698
For a positive integer
n
n
n
, let denote
C
n
C_n
C
n
the figure formed by the inside and perimeter of the circle with center the origin, radius
n
n
n
on the
x
x
x
-
y
y
y
plane.Denote by
N
(
n
)
N(n)
N
(
n
)
the number of a unit square such that all of unit square, whose
x
,
y
x,\ y
x
,
y
coordinates of 4 vertices are integers, and the vertices are included in
C
n
C_n
C
n
.Prove that
lim
n
→
∞
N
(
n
)
n
2
=
π
\lim_{n\to\infty} \frac{N(n)}{n^2}=\pi
lim
n
→
∞
n
2
N
(
n
)
=
π
.
697
1
Hide problems
Today's calculation of Integral 697
Find the volume of the solid of the domain expressed by the inequality
x
2
−
x
≤
y
≤
x
x^2-x\leq y\leq x
x
2
−
x
≤
y
≤
x
, generated by a rotation about the line
y
=
x
.
y=x.
y
=
x
.
696
1
Hide problems
Today's calculation of Integral 696
Let
P
(
x
)
,
Q
(
x
)
P(x),\ Q(x)
P
(
x
)
,
Q
(
x
)
be polynomials such that :
∫
0
2
{
P
(
x
)
}
2
d
x
=
14
,
∫
0
2
P
(
x
)
d
x
=
4
,
∫
0
2
{
Q
(
x
)
}
2
d
x
=
26
,
∫
0
2
Q
(
x
)
d
x
=
2.
\int_0^2 \{P(x)\}^2dx=14,\ \int_0^2 P(x)dx=4,\ \int_0^2 \{Q(x)\}^2dx=26,\ \int_0^2 Q(x)dx=2.
∫
0
2
{
P
(
x
)
}
2
d
x
=
14
,
∫
0
2
P
(
x
)
d
x
=
4
,
∫
0
2
{
Q
(
x
)
}
2
d
x
=
26
,
∫
0
2
Q
(
x
)
d
x
=
2.
Find the maximum and the minimum value of
∫
0
2
P
(
x
)
Q
(
x
)
d
x
\int_0^2 P(x)Q(x)dx
∫
0
2
P
(
x
)
Q
(
x
)
d
x
.
695
1
Hide problems
Today's calculation of Integral 695
For a positive integer
n
n
n
, let
S
n
=
∫
0
1
1
−
(
−
x
)
n
1
+
x
d
x
,
T
n
=
∑
k
=
1
n
(
−
1
)
k
−
1
k
(
k
+
1
)
S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}
S
n
=
∫
0
1
1
+
x
1
−
(
−
x
)
n
d
x
,
T
n
=
k
=
1
∑
n
k
(
k
+
1
)
(
−
1
)
k
−
1
Answer the following questions:(1) Show the following inequality.
∣
S
n
−
∫
0
1
1
1
+
x
d
x
∣
≤
1
n
+
1
\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}
S
n
−
∫
0
1
1
+
x
1
d
x
≤
n
+
1
1
(2) Express
T
n
−
2
S
n
T_n-2S_n
T
n
−
2
S
n
in terms of
n
n
n
.(3) Find the limit
lim
n
→
∞
T
n
.
\lim_{n\to\infty} T_n.
lim
n
→
∞
T
n
.
694
1
Hide problems
Today's calculation of Integral 694
Prove the following inequality:
∫
1
e
(
ln
x
)
2009
x
2
d
x
>
1
2010
⋅
2011
⋅
2012
\int_1^e \frac{(\ln x)^{2009}}{x^2}dx>\frac{1}{2010\cdot 2011\cdot2012}
∫
1
e
x
2
(
ln
x
)
2009
d
x
>
2010
⋅
2011
⋅
2012
1
created by kunny
693
1
Hide problems
Today's calculation of Integral 693
Evaluate
∫
0
π
1
+
∣
cos
x
∣
4
d
x
.
\int_0^{\pi} \sqrt[4]{1+|\cos x|}\ dx.
∫
0
π
4
1
+
∣
cos
x
∣
d
x
.
created by kunny
692
1
Hide problems
Today's calculation of Integral 692
Evaluate
∫
0
π
12
tan
2
x
−
3
3
tan
2
x
−
1
d
x
\int_0^{\frac{\pi}{12}} \frac{\tan ^ 2 x-3}{3\tan ^ 2 x-1}dx
∫
0
12
π
3
t
a
n
2
x
−
1
t
a
n
2
x
−
3
d
x
.created by kunny
691
1
Hide problems
Today's calculation of Integral 691
Let
a
a
a
be a constant. In the
x
y
xy
x
y
palne, the curve
C
1
:
y
=
ln
x
x
C_1:y=\frac{\ln x}{x}
C
1
:
y
=
x
l
n
x
touches
C
2
:
y
=
a
x
2
C_2:y=ax^2
C
2
:
y
=
a
x
2
. Find the volume of the solid generated by a rotation of the part enclosed by
C
1
,
C
2
C_1,\ C_2
C
1
,
C
2
and the
x
x
x
axis about the
x
x
x
axis.2011 Yokohama National Universty entrance exam/Engineering
690
1
Hide problems
Today's calculation of Integral 690
Find the maximum value of
f
(
x
)
=
∫
0
1
t
sin
(
x
+
π
t
)
d
t
f(x)=\int_0^1 t\sin (x+\pi t)\ dt
f
(
x
)
=
∫
0
1
t
sin
(
x
+
π
t
)
d
t
.
689
1
Hide problems
Today's calculation of Integral 689
Let
C
:
y
=
x
2
+
a
x
+
b
C: y=x^2+ax+b
C
:
y
=
x
2
+
a
x
+
b
be a parabola passing through the point
(
1
,
−
1
)
(1,\ -1)
(
1
,
−
1
)
. Find the minimum volume of the figure enclosed by
C
C
C
and the
x
x
x
axis by a rotation about the
x
x
x
axis.Proposed by kunny
688
1
Hide problems
Today's calculation of Integral 688
For a real number
x
x
x
, let
f
(
x
)
=
∫
0
π
2
∣
cos
t
−
x
sin
2
t
∣
d
t
f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt
f
(
x
)
=
∫
0
2
π
∣
cos
t
−
x
sin
2
t
∣
d
t
.(1) Find the minimum value of
f
(
x
)
f(x)
f
(
x
)
.(2) Evaluate
∫
0
1
f
(
x
)
d
x
\int_0^1 f(x)\ dx
∫
0
1
f
(
x
)
d
x
.2011 Tokyo Institute of Technology entrance exam, Problem 2
687
1
Hide problems
Today's calculation of Integral 687
(1) Let
x
>
0
,
y
x>0,\ y
x
>
0
,
y
be real numbers. For variable
t
t
t
, find the difference of Maximum and minimum value of the quadratic function
f
(
t
)
=
x
t
2
+
y
t
f(t)=xt^2+yt
f
(
t
)
=
x
t
2
+
y
t
in
0
≤
t
≤
1
0\leq t\leq 1
0
≤
t
≤
1
.(2) Let
S
S
S
be the domain of the points
(
x
,
y
)
(x,\ y)
(
x
,
y
)
in the coordinate plane forming the following condition:For
x
>
0
x>0
x
>
0
and all real numbers
t
t
t
with
0
≤
t
≤
1
0\leq t\leq 1
0
≤
t
≤
1
, there exists real number
z
z
z
for which
0
≤
x
t
2
+
y
t
+
z
≤
1
0\leq xt^2+yt+z\leq 1
0
≤
x
t
2
+
y
t
+
z
≤
1
.Sketch the outline of
S
S
S
.(3) Let
V
V
V
be the domain of the points
(
x
,
y
,
z
)
(x,\ y,\ z)
(
x
,
y
,
z
)
in the coordinate space forming the following condition:For
0
≤
x
≤
1
0\leq x\leq 1
0
≤
x
≤
1
and for all real numbers
t
t
t
with
0
≤
t
≤
1
0\leq t\leq 1
0
≤
t
≤
1
,
0
≤
x
t
2
+
y
t
+
z
≤
1
0\leq xt^2+yt+z\leq 1
0
≤
x
t
2
+
y
t
+
z
≤
1
holds.Find the volume of
V
V
V
.2011 Tokyo University entrance exam/Science, Problem 6
686
1
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Today's calculation of Integral 686
Let
L
L
L
be a positive constant. For a point
P
(
t
,
0
)
P(t,\ 0)
P
(
t
,
0
)
on the positive part of the
x
x
x
axis on the coordinate plane, denote
Q
(
u
(
t
)
,
v
(
t
)
)
Q(u(t),\ v(t))
Q
(
u
(
t
)
,
v
(
t
))
the point at which the point reach starting from
P
P
P
proceeds by distance
L
L
L
in counter-clockwise on the perimeter of a circle passing the point
P
P
P
with center
O
O
O
. (1) Find
u
(
t
)
,
v
(
t
)
u(t),\ v(t)
u
(
t
)
,
v
(
t
)
.(2) For real number
a
a
a
with
0
<
a
<
1
0<a<1
0
<
a
<
1
, find
f
(
a
)
=
∫
a
1
{
u
′
(
t
)
}
2
+
{
v
′
(
t
)
}
2
d
t
f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt
f
(
a
)
=
∫
a
1
{
u
′
(
t
)
}
2
+
{
v
′
(
t
)
}
2
d
t
.(3) Find
lim
a
→
+
0
f
(
a
)
ln
a
\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}
lim
a
→
+
0
l
n
a
f
(
a
)
.2011 Tokyo University entrance exam/Science, Problem 3
685
1
Hide problems
Today's calculation of Integral 685
Suppose that a cubic function with respect to
x
x
x
,
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
f(x)=ax^3+bx^2+cx+d
f
(
x
)
=
a
x
3
+
b
x
2
+
c
x
+
d
satisfies all of 3 conditions:
f
(
1
)
=
1
,
f
(
−
1
)
=
−
1
,
∫
−
1
1
(
b
x
2
+
c
x
+
d
)
d
x
=
1
f(1)=1,\ f(-1)=-1,\ \int_{-1}^1 (bx^2+cx+d)\ dx=1
f
(
1
)
=
1
,
f
(
−
1
)
=
−
1
,
∫
−
1
1
(
b
x
2
+
c
x
+
d
)
d
x
=
1
.Find
f
(
x
)
f(x)
f
(
x
)
for which
I
=
∫
−
1
1
2
{
f
′
′
(
x
)
}
2
d
x
I=\int_{-1}^{\frac 12} \{f''(x)\}^2\ dx
I
=
∫
−
1
2
1
{
f
′′
(
x
)
}
2
d
x
is minimized, the find the minimum value.2011 Tokyo University entrance exam/Humanities, Problem 1
684
1
Hide problems
Today's calculation of Integral 684
On the
x
y
xy
x
y
plane, find the area of the figure bounded by the graphs of
y
=
x
y=x
y
=
x
and
y
=
∣
3
4
x
2
−
3
∣
−
2
y=\left|\ \frac34 x^2-3\ \right |-2
y
=
4
3
x
2
−
3
−
2
.2011 Kyoto University entrance exam/Science, Problem 3
683
1
Hide problems
Today's calculation of Integral 683
Evaluate
∫
0
1
2
(
x
+
1
)
1
−
2
x
2
d
x
\int_0^{\frac 12} (x+1)\sqrt{1-2x^2}\ dx
∫
0
2
1
(
x
+
1
)
1
−
2
x
2
d
x
.2011 Kyoto University entrance exam/Science, Problem 1B
682
1
Hide problems
Today's calculation of Integral 682
On the
x
x
x
-
y
y
y
plane, 3 half-lines
y
=
0
,
(
x
≥
0
)
,
y
=
x
tan
θ
(
x
≥
0
)
,
y
=
−
3
x
(
x
≤
0
)
y=0,\ (x\geq 0),\ y=x\tan \theta \ (x\geq 0),\ y=-\sqrt{3}x\ (x\leq 0)
y
=
0
,
(
x
≥
0
)
,
y
=
x
tan
θ
(
x
≥
0
)
,
y
=
−
3
x
(
x
≤
0
)
intersect with the circle with the center the origin
O
O
O
, radius
r
≥
1
r\geq 1
r
≥
1
at
A
,
B
,
C
A,\ B,\ C
A
,
B
,
C
respectively. Note that
π
6
≤
θ
≤
π
3
\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}
6
π
≤
θ
≤
3
π
.If the area of quadrilateral
O
A
B
C
OABC
O
A
BC
is one third of the area of the regular hexagon which inscribed in a circle with radius 1, then evaluate
∫
π
6
π
3
r
2
d
θ
.
\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} r^2d\theta .
∫
6
π
3
π
r
2
d
θ
.
2011 Waseda University of Education entrance exam/Science
681
1
Hide problems
Today's calculation of Integral 681
Evaluate
∫
0
π
2
1
−
2
sin
2
x
+
3
cos
2
x
d
x
.
\int_0^{\frac{\pi}{2}} \sqrt{1-2\sin 2x+3\cos ^ 2 x}\ dx.
∫
0
2
π
1
−
2
sin
2
x
+
3
cos
2
x
d
x
.
2011 University of Occupational and Environmental Health/Medicine entrance exam
680
1
Hide problems
Today's calculation of Integral 680
Let
a
>
0
a>0
a
>
0
. Evaluate
∫
0
a
x
2
(
1
−
x
a
)
a
d
x
\int_0^a x^2\left(1-\frac{x}{a}\right)^adx
∫
0
a
x
2
(
1
−
a
x
)
a
d
x
.2011 Keio University entrance exam/Science and Technology
679
1
Hide problems
Today's calculation of Integral 679
Find
∑
k
=
1
3
n
1
∫
0
1
x
(
1
−
x
)
k
d
x
\sum_{k=1}^{3n} \frac{1}{\int_0^1 x(1-x)^k\ dx}
∑
k
=
1
3
n
∫
0
1
x
(
1
−
x
)
k
d
x
1
.2011 Hosei University entrance exam/Design and Enginerring
678
1
Hide problems
Today's calculation of Integral 678
Evaluate
∫
0
π
(
1
+
∑
k
=
1
n
k
cos
k
x
)
2
d
x
(
n
=
1
,
2
,
⋯
)
.
\int_0^{\pi} \left(1+\sum_{k=1}^n k\cos kx\right)^2dx\ \ (n=1,\ 2,\ \cdots).
∫
0
π
(
1
+
k
=
1
∑
n
k
cos
k
x
)
2
d
x
(
n
=
1
,
2
,
⋯
)
.
2011 Doshisya University entrance exam/Life Medical Sciences
677
1
Hide problems
Today's calculation of Integral 677
Let
a
,
b
a,\ b
a
,
b
be positive real numbers with
a
<
b
a<b
a
<
b
. Define the definite integrals
I
1
,
I
2
,
I
3
I_1,\ I_2,\ I_3
I
1
,
I
2
,
I
3
by
I
1
=
∫
a
b
sin
(
x
2
)
d
x
,
I
2
=
∫
a
b
cos
(
x
2
)
x
2
d
x
,
I
3
=
∫
a
b
sin
(
x
2
)
x
4
d
x
I_1=\int_a^b \sin\ (x^2)\ dx,\ I_2=\int_a^b \frac{\cos\ (x^2)}{x^2}\ dx,\ I_3=\int_a^b \frac{\sin\ (x^2)}{x^4}\ dx
I
1
=
∫
a
b
sin
(
x
2
)
d
x
,
I
2
=
∫
a
b
x
2
c
o
s
(
x
2
)
d
x
,
I
3
=
∫
a
b
x
4
s
i
n
(
x
2
)
d
x
.(1) Find the value of
I
1
+
1
2
I
2
I_1+\frac 12I_2
I
1
+
2
1
I
2
in terms of
a
,
b
a,\ b
a
,
b
.(2) Find the value of
I
2
−
3
2
I
3
I_2-\frac 32I_3
I
2
−
2
3
I
3
in terms of
a
,
b
a,\ b
a
,
b
.(3) For a positive integer
n
n
n
, define
K
n
=
∫
2
n
π
2
(
n
+
1
)
π
sin
(
x
2
)
d
x
+
3
4
∫
2
n
π
2
(
n
+
1
)
π
sin
(
x
2
)
x
4
d
x
K_n=\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}} \sin\ (x^2)\ dx+\frac 34\int_{\sqrt{2n\pi}}^{\sqrt{2(n+1)\pi}}\frac{\sin\ (x^2)}{x^4}\ dx
K
n
=
∫
2
nπ
2
(
n
+
1
)
π
sin
(
x
2
)
d
x
+
4
3
∫
2
nπ
2
(
n
+
1
)
π
x
4
s
i
n
(
x
2
)
d
x
.Find the value of
lim
n
→
∞
2
n
π
2
n
π
K
n
\lim_{n\to\infty} 2n\pi \sqrt{2n\pi} K_n
lim
n
→
∞
2
nπ
2
nπ
K
n
.2011 Tokyo University of Science entrance exam/Information Sciences, Applied Chemistry, Mechanical Enginerring, Civil Enginerring
676
1
Hide problems
Today's calculation of Integral 676
Let
f
(
x
)
=
cos
4
x
+
3
sin
4
x
f(x)=\cos ^ 4 x+3\sin ^ 4 x
f
(
x
)
=
cos
4
x
+
3
sin
4
x
. Evaluate
∫
0
π
2
∣
f
′
(
x
)
∣
d
x
\int_0^{\frac{\pi}{2}} |f'(x)|dx
∫
0
2
π
∣
f
′
(
x
)
∣
d
x
.2011 Tokyo University of Science entrance exam/Management
675
1
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Today's calculation of Integral 675
In the coordinate plane with the origin
O
O
O
, consider points
P
(
t
+
2
,
0
)
,
Q
(
0
,
−
2
t
2
−
2
t
+
4
)
(
t
≥
0
)
.
P(t+2,\ 0),\ Q(0, -2t^2-2t+4)\ (t\geq 0).
P
(
t
+
2
,
0
)
,
Q
(
0
,
−
2
t
2
−
2
t
+
4
)
(
t
≥
0
)
.
If the
y
y
y
-coordinate of
Q
Q
Q
is nonnegative, then find the area of the region swept out by the line segment
P
Q
PQ
PQ
.2011 Ritsumeikan University entrance exam/Pharmacy
674
1
Hide problems
Today's calculation of Integral 674
Evaluate
∫
0
1
x
2
+
5
(
x
+
1
)
2
(
x
−
2
)
d
x
.
\int_0^1 \frac{x^2+5}{(x+1)^2(x-2)}dx.
∫
0
1
(
x
+
1
)
2
(
x
−
2
)
x
2
+
5
d
x
.
2011 Doshisya University entrance exam/Science and Technology
673
1
Hide problems
Today's calculation of Integral 673
Let
f
(
x
)
=
∫
0
x
1
1
+
t
2
d
t
.
f(x)=\int_0^ x \frac{1}{1+t^2}dt.
f
(
x
)
=
∫
0
x
1
+
t
2
1
d
t
.
For
−
1
≤
x
<
1
-1\leq x<1
−
1
≤
x
<
1
, find
cos
{
2
f
(
1
+
x
1
−
x
)
}
.
\cos \left\{2f\left(\sqrt{\frac{1+x}{1-x}}\right)\right\}.
cos
{
2
f
(
1
−
x
1
+
x
)
}
.
2011 Ritsumeikan University entrance exam/Science and Technology
709
1
Hide problems
Today's calculation of Integral 709
Evaluate \int_0^1 \frac{x}{1\plus{}x}\sqrt{1\minus{}x^2}\ dx.
708
1
Hide problems
Today's calculation of Integral 708
Find \lim_{n\to\infty} \int_0^1 x^2|\sin n\pi x|\ dx\ (n\equal{}1,\ 2,\cdots).