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Today's Calculation Of Integral
2006 Today's Calculation Of Integral
2006 Today's Calculation Of Integral
Part of
Today's Calculation Of Integral
Subcontests
(76)
149
1
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Today's calculation of Integral 149
Let
f
(
x
)
=
(
1
−
x
2
)
3
2
f(x)=(1-x^2)^{\frac 32}
f
(
x
)
=
(
1
−
x
2
)
2
3
. Denote
M
M
M
the maximum value of
∣
f
′
(
x
)
∣
|f'(x)|
∣
f
′
(
x
)
∣
in
(
−
1
,
1
)
(-1,\ 1)
(
−
1
,
1
)
. Prove that
∫
−
1
1
f
(
x
)
d
x
≤
M
.
\int_{-1}^1 f(x)\ dx\leq M.
∫
−
1
1
f
(
x
)
d
x
≤
M
.
1981 Musashi Institute of Technology entrance exam/Electro Telecommunication, Civil Enginerring
167
1
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Today's calculation of Integral 167
In
x
y
z
xyz
x
yz
plane find the volume of the solid formed by the points
(
x
,
y
,
z
)
(x,\ y,\ z)
(
x
,
y
,
z
)
satisfying the following system of inequalities.
0
≤
z
≤
1
+
x
+
y
−
3
(
x
−
y
)
y
,
0
≤
y
≤
1
,
y
≤
x
≤
y
+
1.
0\leq z\leq 1+x+y-3(x-y)y,\ 0\leq y\leq 1,\ y\leq x\leq y+1.
0
≤
z
≤
1
+
x
+
y
−
3
(
x
−
y
)
y
,
0
≤
y
≤
1
,
y
≤
x
≤
y
+
1.
166
1
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Today's calculation of Integral 166
Express the following the limit values in terms of a definite integral and find them. (1)
I
=
lim
n
→
∞
1
n
ln
(
1
+
1
n
)
(
1
+
2
n
)
⋯
⋯
(
1
+
n
n
)
.
I=\lim_{n\to\infty}\frac{1}{n}\ln \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\cdots\cdots \left(1+\frac{n}{n}\right).
I
=
lim
n
→
∞
n
1
ln
(
1
+
n
1
)
(
1
+
n
2
)
⋯⋯
(
1
+
n
n
)
.
(2)
J
=
lim
n
→
∞
1
n
2
(
n
2
−
1
+
n
2
−
2
2
+
⋯
⋯
+
n
2
−
n
2
)
.
J=\lim_{n\to\infty}\frac{1}{n^{2}}(\sqrt{n^{2}-1}+\sqrt{n^{2}-2^{2}}+\cdots\cdots+\sqrt{n^{2}-n^{2}}).
J
=
lim
n
→
∞
n
2
1
(
n
2
−
1
+
n
2
−
2
2
+
⋯⋯
+
n
2
−
n
2
)
.
(3)
K
=
lim
n
→
∞
1
n
3
(
n
2
+
1
+
2
n
2
+
2
2
+
⋯
+
n
n
2
+
n
2
)
.
K=\lim_{n\to\infty}\frac{1}{n^{3}}(\sqrt{n^{2}+1}+2\sqrt{n^{2}+2^{2}}+\cdots+n\sqrt{n^{2}+n^{2}}).
K
=
lim
n
→
∞
n
3
1
(
n
2
+
1
+
2
n
2
+
2
2
+
⋯
+
n
n
2
+
n
2
)
.
165
1
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Today's calculation of Integral 165
On
x
−
y
x-y
x
−
y
plane, let
C
:
y
=
2006
x
3
−
12070102
x
2
+
⋯
.
C: y=2006x^{3}-12070102x^{2}+\cdots.
C
:
y
=
2006
x
3
−
12070102
x
2
+
⋯
.
Find the area of the region surrounded by the tangent line of
C
C
C
at
x
=
2006
x=2006
x
=
2006
and the curve
C
.
C.
C
.
164
1
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Today's calculation of Integral 164
For positive integers
n
,
n,
n
,
let
S
n
=
1
1
+
1
2
+
⋯
⋯
+
1
n
,
T
n
=
1
1
+
1
2
+
1
2
+
1
2
+
⋯
⋯
+
1
n
+
1
2
.
S_{n}=\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\cdots\cdots+\frac{1}{\sqrt{n}},\ T_{n}=\frac{1}{\sqrt{1+\frac{1}{2}}}+\frac{1}{\sqrt{2+\frac{1}{2}}}+\cdots\cdots+\frac{1}{\sqrt{n+\frac{1}{2}}}.
S
n
=
1
1
+
2
1
+
⋯⋯
+
n
1
,
T
n
=
1
+
2
1
1
+
2
+
2
1
1
+
⋯⋯
+
n
+
2
1
1
.
Find
lim
n
→
∞
T
n
S
n
.
\lim_{n\to\infty}\frac{T_{n}}{S_{n}}.
lim
n
→
∞
S
n
T
n
.
163
1
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Today's calculation of Integral 163
Let
I
n
=
∫
0
π
4
tan
n
x
d
x
(
n
=
0
,
1
,
2
,
⋯
)
.
I_{n}=\int_{0}^{\frac{\pi}{4}}\tan^{n}x\ dx\ (n=0,\ 1,\ 2,\ \cdots).
I
n
=
∫
0
4
π
tan
n
x
d
x
(
n
=
0
,
1
,
2
,
⋯
)
.
Find
∑
n
=
0
∞
{
I
n
+
2
2
+
(
I
n
+
1
+
I
n
+
3
)
I
n
+
2
+
I
n
+
1
I
n
+
3
}
.
\sum_{n=0}^{\infty}\{{I_{n+2}}^{2}+(I_{n+1}+I_{n+3})I_{n+2}+I_{n+1}I_{n+3}\}.
∑
n
=
0
∞
{
I
n
+
2
2
+
(
I
n
+
1
+
I
n
+
3
)
I
n
+
2
+
I
n
+
1
I
n
+
3
}
.
162
1
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Today's calculation of Integral 162
Let
f
(
x
)
f(x)
f
(
x
)
be the function such that
f
(
x
)
>
0
f(x)>0
f
(
x
)
>
0
at
x
≥
0
x\geq 0
x
≥
0
and
{
f
(
x
)
}
2006
=
∫
0
x
f
(
t
)
d
t
+
1.
\{f(x)\}^{2006}=\int_{0}^{x}f(t) dt+1.
{
f
(
x
)
}
2006
=
∫
0
x
f
(
t
)
d
t
+
1.
Find the value of
{
f
(
2006
)
}
2005
.
\{f(2006)\}^{2005}.
{
f
(
2006
)
}
2005
.
160
1
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Today's calculation of Integral 160
Find the value of
m
(
0
<
m
<
1
)
m\ (0<m<1)
m
(
0
<
m
<
1
)
for which
∫
0
π
∣
sin
x
−
m
x
∣
d
x
\int_{0}^{\pi}|\sin x-mx|\ dx
∫
0
π
∣
sin
x
−
m
x
∣
d
x
is minimized.
161
1
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Today's calculation of Integral 161
Find the differentiable function
f
(
x
)
f(x)
f
(
x
)
such that
f
(
x
)
=
−
∫
0
x
f
(
t
)
tan
t
d
t
+
∫
0
x
tan
(
t
−
x
)
d
t
(
∣
x
∣
<
π
2
)
.
f(x)=-\int_{0}^{x}f(t)\tan t\ dt+\int_{0}^{x}\tan (t-x)\ dt\ \left(|x|<\frac{\pi}{2}\right).
f
(
x
)
=
−
∫
0
x
f
(
t
)
tan
t
d
t
+
∫
0
x
tan
(
t
−
x
)
d
t
(
∣
x
∣
<
2
π
)
.
159
1
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Today's calculation of Integral 159
A function is defined by
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
.
(1) Find the equation of normal line at
x
=
1
x=1
x
=
1
of
y
=
f
(
x
)
.
y=f(x).
y
=
f
(
x
)
.
(2) Find the area of the figure surrounded by the normal line found in (1), the
x
x
x
axis and the graph of
y
=
f
(
x
)
.
y=f(x).
y
=
f
(
x
)
.
Note that you may not use the formula
∫
1
1
+
x
2
d
x
=
tan
−
1
x
+
C
o
n
s
t
.
\int \frac{1}{1+x^{2}}dx=\tan^{-1}x+Const.
∫
1
+
x
2
1
d
x
=
tan
−
1
x
+
C
o
n
s
t
.
158
1
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Today's calculation of Integral 158
(1) Evaluate the definite integral
∫
0
π
e
−
x
sin
x
d
x
.
\int_{0}^{\pi}e^{-x}\sin x dx.
∫
0
π
e
−
x
sin
x
d
x
.
(2) Find the limit
lim
n
→
∞
∫
0
n
π
e
−
x
∣
sin
x
∣
d
x
.
\lim_{n\to\infty}\int_{0}^{n\pi}e^{-x}|\sin x| dx.
lim
n
→
∞
∫
0
nπ
e
−
x
∣
sin
x
∣
d
x
.
157
1
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Today's calculation of Integral 157
Find the volume of the solid expressed by the following six inequaities in
x
y
z
xyz
x
yz
space.
x
≥
0
,
y
≥
0
,
z
≥
0
,
x
+
y
+
z
≤
3
,
x
+
2
z
≤
4
,
y
−
z
≤
1.
x\geq 0,\ y\geq 0,\ z\geq 0,\ x+y+z\leq 3,\ x+2z\leq 4,\ y-z\leq 1.
x
≥
0
,
y
≥
0
,
z
≥
0
,
x
+
y
+
z
≤
3
,
x
+
2
z
≤
4
,
y
−
z
≤
1.
156
1
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Today's calculation of Integral 156
For arbiterary integers
n
,
n,
n
,
find the continuous function
f
(
x
)
f(x)
f
(
x
)
which satisfies the following equation.
lim
h
→
0
1
h
∫
x
−
n
h
x
+
n
h
f
(
t
)
d
t
=
2
f
(
n
x
)
.
\lim_{h\rightarrow 0}\frac{1}{h}\int_{x-nh}^{x+nh}f(t) dt=2f(nx).
h
→
0
lim
h
1
∫
x
−
nh
x
+
nh
f
(
t
)
d
t
=
2
f
(
n
x
)
.
Note that
x
x
x
can range all real numbers and
f
(
1
)
=
1.
f(1)=1.
f
(
1
)
=
1.
155
1
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Today's calculation of Integral 155
The sequence
{
c
n
}
\{c_{n}\}
{
c
n
}
is determined by the following equation.
c
n
=
(
n
+
1
)
∫
0
1
x
n
cos
π
x
d
x
(
n
=
1
,
2
,
⋯
)
.
c_{n}=(n+1)\int_{0}^{1}x^{n}\cos \pi x\ dx\ (n=1,\ 2,\ \cdots).
c
n
=
(
n
+
1
)
∫
0
1
x
n
cos
π
x
d
x
(
n
=
1
,
2
,
⋯
)
.
Let
λ
\lambda
λ
be the limit value
lim
n
→
∞
c
n
.
\lim_{n\to\infty}c_{n}.
lim
n
→
∞
c
n
.
Find
lim
n
→
∞
c
n
+
1
−
λ
c
n
−
λ
.
\lim_{n\to\infty}\frac{c_{n+1}-\lambda}{c_{n}-\lambda}.
lim
n
→
∞
c
n
−
λ
c
n
+
1
−
λ
.
154
1
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Today's calculation of Integral 154
Find the function
f
(
x
)
f(x)
f
(
x
)
which is defined for
[
−
π
2
,
π
2
]
\left[-\frac{\pi}{2},\ \frac{\pi}{2}\right]
[
−
2
π
,
2
π
]
such that
f
(
x
)
+
∫
−
π
2
π
2
sin
(
x
−
y
)
⋅
f
(
y
)
d
y
=
x
+
1
(
−
π
2
≤
x
≤
π
2
)
.
f(x)+\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sin (x-y)\cdot f(y)\ dy=x+1\ \left(-\frac{\pi}{2}\leq x\leq \frac{\pi}{2}\right).
f
(
x
)
+
∫
−
2
π
2
π
sin
(
x
−
y
)
⋅
f
(
y
)
d
y
=
x
+
1
(
−
2
π
≤
x
≤
2
π
)
.
153
1
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Today's calculation of Integral 153
Draw the perpendicular to the tangent line of the ellipse
x
2
a
2
+
y
2
b
2
=
1
(
a
>
0
,
b
>
0
)
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\ (a>0,\ b>0)
a
2
x
2
+
b
2
y
2
=
1
(
a
>
0
,
b
>
0
)
from the origin
O
(
0
,
0
)
.
O(0,\ 0).
O
(
0
,
0
)
.
Let
θ
\theta
θ
be the angle between the perpendicular and the positive direction of
x
x
x
axis. Denote the length of the perpendicular by
r
(
θ
)
.
r(\theta).
r
(
θ
)
.
Calculate
∫
0
2
π
r
(
θ
)
2
d
θ
.
\int_{0}^{2\pi}r(\theta )^{2}\ d\theta.
∫
0
2
π
r
(
θ
)
2
d
θ
.
152
1
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Today's calculation of Integral 152
Let
f
(
x
)
f(x)
f
(
x
)
the function such that
f
(
0
)
=
0
,
∣
f
′
(
x
)
∣
≤
1
1
+
x
(
x
≥
0
)
.
f(0)=0,\ |f'(x)|\leq \frac{1}{1+x}\ (x\geq 0).
f
(
0
)
=
0
,
∣
f
′
(
x
)
∣
≤
1
+
x
1
(
x
≥
0
)
.
Prove that
∫
0
e
−
1
{
f
(
x
)
}
2
d
x
≤
e
−
2.
\int_{0}^{e-1}\{f(x)\}^{2}dx\leq e-2.
∫
0
e
−
1
{
f
(
x
)
}
2
d
x
≤
e
−
2.
151
1
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Today's calculation of Integral 151
Let
a
,
b
a,\ b
a
,
b
be positive constant numbers.Find the volume of the revolution of the region surrounded by the parabola
y
=
a
x
2
y=ax^{2}
y
=
a
x
2
and the line
y
=
b
x
y=bx
y
=
b
x
about the line
y
=
b
x
y=bx
y
=
b
x
as the axis on
x
y
xy
x
y
plane.
150
1
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Today's calculation of Integral 150
Find the value of
a
a
a
such that
lim
n
→
∞
3
2
∫
−
a
3
a
3
(
1
−
t
3
n
)
n
t
2
d
t
=
2
(
n
=
1
,
2
,
⋯
)
.
\lim_{n\to\infty}\frac{3}{2}\int_{-\sqrt[3]{a}}^{\sqrt[3]{a}}\left(1-\frac{t^{3}}{n}\right)^{n}t^{2}\ dt=\sqrt{2}\ \ (n=1,2,\cdots).
lim
n
→
∞
2
3
∫
−
3
a
3
a
(
1
−
n
t
3
)
n
t
2
d
t
=
2
(
n
=
1
,
2
,
⋯
)
.
148
1
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Today's calculation of Integral 148
Evaluate
∫
0
π
2
sin
2
n
θ
sin
2
θ
d
θ
(
n
=
1
,
2
,
⋯
)
.
\int_{0}^{\frac{\pi}{2}}\frac{\sin^{2}n\theta}{\sin^{2}\theta}\ d\theta\ (n=1,\ 2,\ \cdots).
∫
0
2
π
s
i
n
2
θ
s
i
n
2
n
θ
d
θ
(
n
=
1
,
2
,
⋯
)
.
147
1
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Today's calculation of Integral 147
Find the area of the figure surrounded by the curve
2
(
x
2
+
1
)
y
2
+
8
x
2
y
+
x
4
+
4
x
2
−
1
=
0.
2(x^{2}+1)y^{2}+8x^{2}y+x^{4}+4x^{2}-1=0.
2
(
x
2
+
1
)
y
2
+
8
x
2
y
+
x
4
+
4
x
2
−
1
=
0.
146
1
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Today's calculation of Integral 146
Find the maximum value of
∫
−
1
1
∣
x
−
a
∣
e
x
d
x
\int_{-1}^{1}|x-a|e^{x}dx
∫
−
1
1
∣
x
−
a
∣
e
x
d
x
for
∣
a
∣
≤
1.
|a|\leq 1.
∣
a
∣
≤
1.
145
1
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Today's calculation of Integral 145
Find the minimum value of
∫
x
x
+
l
(
t
+
1
t
)
d
t
(
x
>
0
,
l
>
0
)
.
\int_{x}^{x+l}\left(t+\frac{1}{t}\right)dt \ (x>0,\ l>0).
∫
x
x
+
l
(
t
+
t
1
)
d
t
(
x
>
0
,
l
>
0
)
.
144
1
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Today's calculation of Integral 144
Evaluate
lim
n
→
∞
∫
0
π
∣
(
x
+
π
n
)
sin
n
x
∣
d
x
(
n
=
1
,
2
,
⋯
)
.
\lim_{n\to\infty}\int_{0}^\pi \left|\left(x+\frac{\pi}{n}\right)\sin nx\right|\ dx\ (n=1,\ 2,\ \cdots).
lim
n
→
∞
∫
0
π
(
x
+
n
π
)
sin
n
x
d
x
(
n
=
1
,
2
,
⋯
)
.
143
1
Hide problems
Today's calculation of Integral 143
Evaluate
∫
0
π
2
1
−
sin
2
x
(
1
+
sin
2
x
)
2
d
x
.
\int_{0}^{\frac{\pi}{2}}\frac{1-\sin 2x}{(1+\sin 2x)^{2}}\ dx.
∫
0
2
π
(
1
+
s
i
n
2
x
)
2
1
−
s
i
n
2
x
d
x
.
142
1
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Today's calculation of Integral 142
Evaluate
∫
0
π
sin
x
1
−
2
a
cos
x
+
a
2
d
x
(
a
>
0
)
.
\int_{0}^\pi \frac{\sin x}{\sqrt{1-2a\cos x+a^{2}}}\ dx\ (a>0).
∫
0
π
1
−
2
a
c
o
s
x
+
a
2
s
i
n
x
d
x
(
a
>
0
)
.
141
1
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Today's calculation of Integral 141
Evaluate
∫
0
π
cos
4
x
−
cos
4
α
cos
x
−
cos
α
d
x
.
\int_{0}^{\pi}\frac{\cos 4x-\cos 4\alpha}{\cos x-\cos \alpha}\ dx.
∫
0
π
c
o
s
x
−
c
o
s
α
c
o
s
4
x
−
c
o
s
4
α
d
x
.
140
1
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Today's calculation of Integral 140
Evaluate
∫
0
π
4
(
cos
x
sin
x
+
cos
x
)
2
d
x
,
∫
0
π
4
(
sin
x
+
cos
x
cos
x
)
2
d
x
.
\int_{0}^{\frac{\pi}{4}}\left(\frac{\cos x}{\sin x+\cos x}\right)^{2}\ dx,\ \ \ \int_{0}^{\frac{\pi}{4}}\left(\frac{\sin x+\cos x}{\cos x}\right)^{2}\ dx.
∫
0
4
π
(
sin
x
+
cos
x
cos
x
)
2
d
x
,
∫
0
4
π
(
cos
x
sin
x
+
cos
x
)
2
d
x
.
139
1
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Today's calculation of Integral 139
Let
a
,
b
a,\ b
a
,
b
be real numbers. Evaluate
∫
0
2
π
(
a
cos
x
+
b
sin
x
)
2
n
d
x
(
n
=
1
,
2
,
⋯
)
.
\int_{0}^{2\pi}(a\cos x+b\sin x)^{2n}\ dx\ (n=1,\ 2,\ \cdots).
∫
0
2
π
(
a
cos
x
+
b
sin
x
)
2
n
d
x
(
n
=
1
,
2
,
⋯
)
.
138
1
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Today's calculation of Integral 138
Let
f
(
x
)
f(x)
f
(
x
)
be the product of functions made by taking four functions from three functions
x
,
sin
x
,
cos
x
x,\ \sin x,\ \cos x
x
,
sin
x
,
cos
x
repeatedly. Find the minimum value of
∫
0
π
2
f
(
x
)
d
x
.
\int_{0}^{\frac{\pi}{2}}f(x)\ dx.
∫
0
2
π
f
(
x
)
d
x
.
137
1
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Today's calculation of Integral 137
Find the value of
a
a
a
for which
∫
0
1
∣
x
e
x
−
a
∣
d
x
\int_{0}^{1}|xe^{x}-a|\ dx
∫
0
1
∣
x
e
x
−
a
∣
d
x
is minimized.
136
1
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Today's calculation of Integral 136
Let
c
c
c
be the constant number such that
c
>
1.
c>1.
c
>
1.
Find the least area of the figure surrounded by the line passing through the point
(
1
,
c
)
(1,\ c)
(
1
,
c
)
and the palabola
y
=
x
2
y=x^{2}
y
=
x
2
on
x
−
y
x-y
x
−
y
plane.
135
1
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Today's calculation of Integral 135
Find the value of
a
a
a
for which
∫
0
π
2
∣
a
sin
x
−
cos
x
∣
d
x
(
a
>
0
)
\int_{0}^{\frac{\pi}{2}}|a\sin x-\cos x|\ dx\ (a>0)
∫
0
2
π
∣
a
sin
x
−
cos
x
∣
d
x
(
a
>
0
)
is minimized.
134
1
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Today's calculation of Integral 134
For positive integers
n
,
n,
n
,
let
A
n
=
1
n
{
(
n
+
1
)
+
(
n
+
2
)
+
⋯
+
(
n
+
n
)
}
,
B
n
=
{
(
n
+
1
)
(
n
+
2
)
⋯
(
n
+
n
)
}
1
n
.
A_{n}=\frac{1}{n}\{(n+1)+(n+2)+\cdots+(n+n)\},\ B_{n}=\{(n+1)(n+2)\cdots (n+n)\}^{\frac{1}{n}}.
A
n
=
n
1
{(
n
+
1
)
+
(
n
+
2
)
+
⋯
+
(
n
+
n
)}
,
B
n
=
{(
n
+
1
)
(
n
+
2
)
⋯
(
n
+
n
)
}
n
1
.
Find
lim
n
→
∞
A
n
B
n
.
\lim_{n\to\infty}\frac{A_{n}}{B_{n}}.
lim
n
→
∞
B
n
A
n
.
133
1
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Today's calculation of Integral 133
Let
f
(
x
)
f(x)
f
(
x
)
be the polynomial with respect to
x
,
x,
x
,
and
g
n
(
x
)
=
n
−
2
n
2
∣
x
−
1
2
∣
+
∣
n
−
2
n
2
∣
x
−
1
2
∣
∣
.
g_{n}(x)=n-2n^{2}\left|x-\frac{1}{2}\right|+\left|n-2n^{2}\left|x-\frac{1}{2}\right|\right|.
g
n
(
x
)
=
n
−
2
n
2
x
−
2
1
+
n
−
2
n
2
x
−
2
1
.
Find
lim
n
→
∞
∫
0
1
f
(
x
)
g
n
(
x
)
d
x
.
\lim_{n\to\infty}\int_{0}^{1}f(x)g_{n}(x)\ dx.
lim
n
→
∞
∫
0
1
f
(
x
)
g
n
(
x
)
d
x
.
132
1
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Today's calculation of Integral 132
Find the area of the figure such that the points
(
x
,
y
)
(x,\ y)
(
x
,
y
)
satisfies the inequality
lim
n
→
∞
(
x
2
n
+
y
2
n
)
1
n
≥
3
2
x
2
+
3
2
y
2
−
1.
\lim_{n\to\infty}(x^{2n}+y^{2n})^{\frac{1}{n}}\geq \frac{3}{2}x^{2}+\frac{3}{2}y^{2}-1.
lim
n
→
∞
(
x
2
n
+
y
2
n
)
n
1
≥
2
3
x
2
+
2
3
y
2
−
1.
131
1
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Today's calculation of Integral 131
For
a
>
0
,
a>0,
a
>
0
,
find the minimum value of
∫
0
1
a
(
a
3
+
4
x
−
a
5
x
2
)
e
a
x
d
x
.
\int_{0}^{\frac{1}{a}}(a^{3}+4x-a^{5}x^{2})e^{ax}\ dx.
∫
0
a
1
(
a
3
+
4
x
−
a
5
x
2
)
e
a
x
d
x
.
130
1
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Today's calculation of Integral 130
Find the value of
a
a
a
such that
∫
0
a
1
e
x
+
4
e
−
x
+
5
d
x
=
ln
2
3
.
\int_{0}^{a}\frac{1}{e^{x}+4e^{-x}+5}\ dx=\ln \sqrt[3]{2}.
∫
0
a
e
x
+
4
e
−
x
+
5
1
d
x
=
ln
3
2
.
129
1
Hide problems
Today's calculation of Integral 129
The sequence
{
a
n
}
\{a_{n}\}
{
a
n
}
is defined as follows.
a
1
=
π
4
,
a
n
=
∫
0
1
2
(
cos
π
x
+
a
n
−
1
)
cos
π
x
d
x
(
n
=
2
,
3
,
⋯
)
a_{1}=\frac{\pi}{4},\ a_{n}=\int_{0}^{\frac{1}{2}}(\cos \pi x+a_{n-1})\cos \pi x\ dx\ \ (n=2,3,\cdots)
a
1
=
4
π
,
a
n
=
∫
0
2
1
(
cos
π
x
+
a
n
−
1
)
cos
π
x
d
x
(
n
=
2
,
3
,
⋯
)
Find
lim
n
→
∞
a
n
\lim_{n\to\infty}a_{n}
lim
n
→
∞
a
n
.
128
1
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Today's calculation of Integral 128
Prove the following inequality.
−
π
3
ln
2
+
π
3
81
<
∫
0
π
3
ln
(
cos
x
)
d
x
<
−
π
3
162
.
-\frac{\pi}{3}\ln 2+\frac{\pi^{3}}{81}<\int_{0}^{\frac{\pi}{3}}\ln (\cos x) dx<-\frac{\pi^{3}}{162}.
−
3
π
ln
2
+
81
π
3
<
∫
0
3
π
ln
(
cos
x
)
d
x
<
−
162
π
3
.
127
1
Hide problems
Today's calculation of Integral 127
The problem was didn't make sense, so that was deleted.
126
1
Hide problems
Today's calculation of Integral 126
For
t
>
0
,
t>0,
t
>
0
,
find the minimum value of
∫
0
1
x
∣
e
−
x
2
−
t
∣
d
x
.
\int_{0}^{1}x|e^{-x^{2}}-t| dx.
∫
0
1
x
∣
e
−
x
2
−
t
∣
d
x
.
125
1
Hide problems
Today's calculation of Integral 125
Prove the following inequality for
x
≥
0.
x\geq 0.
x
≥
0.
∫
0
x
(
t
−
t
2
)
sin
2004
t
d
t
<
1
2006
\int_{0}^{x}(t-t^{2})\sin^{2004}t\ dt <\frac{1}{2006}
∫
0
x
(
t
−
t
2
)
sin
2004
t
d
t
<
2006
1
124
1
Hide problems
Today's calculation of Integral 124
Let
a
>
1.
a>1.
a
>
1.
Find the area
S
(
a
)
S(a)
S
(
a
)
of the part surrounded by the curve
y
=
a
4
(
a
2
−
x
2
)
3
(
0
≤
x
≤
1
)
,
x
y=\frac{a^{4}}{\sqrt{(a^{2}-x^{2})^{3}}}\ (0\leq x\leq 1),\ x
y
=
(
a
2
−
x
2
)
3
a
4
(
0
≤
x
≤
1
)
,
x
axis ,
y
y
y
axis and the line
x
=
1
,
x=1,
x
=
1
,
then when
a
a
a
varies in the range of
a
>
1
,
a>1,
a
>
1
,
then find the extremal value of
S
(
a
)
.
S(a).
S
(
a
)
.
123
1
Hide problems
Today's calculation of Integral 123
Let
f
(
x
)
=
π
x
2
sin
π
x
2
.
f(x)=\pi x^{2}\sin \pi x^{2}.
f
(
x
)
=
π
x
2
sin
π
x
2
.
Prove that the volume
V
V
V
formed by the revolution of the figure surrounded by the part
0
≤
x
≤
1
0\leq x\leq 1
0
≤
x
≤
1
of the graph of
y
=
f
(
x
)
y=f(x)
y
=
f
(
x
)
and
x
x
x
axis about
y
y
y
axis is can be given as
2
π
∫
0
1
x
f
(
x
)
d
x
2\pi \int_{0}^{1}xf(x)\ dx
2
π
∫
0
1
x
f
(
x
)
d
x
then find the value of
V
.
V.
V
.
122
1
Hide problems
Today's calculation of Integral 122
Let
x
(
t
)
=
tan
t
,
y
(
t
)
=
−
ln
cos
t
(
−
π
2
<
t
<
π
2
)
.
x(t)=\tan t ,\ y(t)=-\ln \cos t\ \left(-\frac{\pi}{2}<t<\frac{\pi}{2}\right).
x
(
t
)
=
tan
t
,
y
(
t
)
=
−
ln
cos
t
(
−
2
π
<
t
<
2
π
)
.
Find the area surrounded by the curve
:
x
=
x
(
t
)
,
y
=
y
(
t
)
: x=x(t),\ y=y(t)
:
x
=
x
(
t
)
,
y
=
y
(
t
)
and
x
x
x
axis and the line
x
=
1.
x=1.
x
=
1.
121
1
Hide problems
Today's calculation of Integral 121
Given the parabola
C
:
y
=
x
2
.
C: y=x^{2}.
C
:
y
=
x
2
.
If the circle centered at
y
y
y
axis with radius
1
1
1
has common tangent lines with
C
C
C
at distinct two points, then find the coordinate of the center of the circle
K
K
K
and the area of the figure surrounded by
C
C
C
and
K
.
K.
K
.
120
1
Hide problems
Today's calculation of Integral 120
Let
k
k
k
be real constants.How many real roots can the following quadratic equation have?
x
2
=
−
2
x
+
k
+
∫
0
1
∣
t
+
k
∣
d
t
.
x^{2}=-2x+k+\int_{0}^{1}|t+k|\ dt.
x
2
=
−
2
x
+
k
+
∫
0
1
∣
t
+
k
∣
d
t
.
119
1
Hide problems
Today's calculation of Integral 119
Find the continuous function
f
(
x
)
f(x)
f
(
x
)
and constant number such that
∫
0
x
f
(
t
)
d
t
=
e
x
−
a
e
2
x
∫
0
1
f
(
t
)
e
−
t
d
t
.
\int_{0}^{x}f(t)\ dt=e^{x}-ae^{2x}\int_{0}^{1}f(t)e^{-t}\ dt.
∫
0
x
f
(
t
)
d
t
=
e
x
−
a
e
2
x
∫
0
1
f
(
t
)
e
−
t
d
t
.
118
1
Hide problems
Today's calculation of Integral 118
Let
f
(
x
)
f(x)
f
(
x
)
be the function defined for
x
≥
0
x\geq 0
x
≥
0
which satisfies the following conditions. (a)
f
(
x
)
=
{
x
(
0
≤
x
<
1
)
2
−
x
(
1
≤
x
<
2
)
f(x)=\begin{cases}x \ \ \ \ \ \ \ \ ( 0\leq x<1) \\ 2-x \ \ \ (1\leq x <2) \end{cases}
f
(
x
)
=
{
x
(
0
≤
x
<
1
)
2
−
x
(
1
≤
x
<
2
)
(b)
f
(
x
+
2
n
)
=
f
(
x
)
(
n
=
1
,
2
,
⋯
)
f(x+2n)=f(x) \ (n=1,2,\cdots)
f
(
x
+
2
n
)
=
f
(
x
)
(
n
=
1
,
2
,
⋯
)
Find
lim
n
→
∞
∫
0
2
n
f
(
x
)
e
−
x
d
x
.
\lim_{n\to\infty}\int_{0}^{2n}f(x)e^{-x}\ dx.
lim
n
→
∞
∫
0
2
n
f
(
x
)
e
−
x
d
x
.
117
1
Hide problems
Today's calculation of Integral 117
Let
a
a
a
be a real constant number. Evaluate
lim
n
→
∞
n
∫
−
1
1
e
−
n
∣
a
−
x
∣
d
x
.
\lim_{n\to\infty}n\int_{-1}^{1}e^{-n|a-x|}\ dx.
lim
n
→
∞
n
∫
−
1
1
e
−
n
∣
a
−
x
∣
d
x
.
116
1
Hide problems
Today's calculation of Integral 116
Find
lim
t
→
0
∫
0
2
π
∣
sin
(
x
+
t
)
−
sin
x
∣
d
x
∣
t
∣
.
\lim_{t\rightarrow 0}\int_{0}^{2\pi}\frac{|\sin (x+t)-\sin x|\ dx}{|t|}.
lim
t
→
0
∫
0
2
π
∣
t
∣
∣
s
i
n
(
x
+
t
)
−
s
i
n
x
∣
d
x
.
115
1
Hide problems
Today's calculation of Integral 115
Find the value of
a
a
a
such that
∫
0
π
2
(
sin
x
+
a
cos
x
)
3
d
x
−
4
a
π
−
2
∫
0
π
2
x
cos
x
d
x
=
2.
\int_{0}^\frac{\pi}{2}(\sin x+a\cos x)^{3}\ dx-\frac{4a}{\pi-2}\int_{0}^{\frac{\pi}{2}}x\cos x dx=2.
∫
0
2
π
(
sin
x
+
a
cos
x
)
3
d
x
−
π
−
2
4
a
∫
0
2
π
x
cos
x
d
x
=
2.
114
1
Hide problems
Today's calculation of Integral 114
Let
a
a
a
be positive numbers.For
∣
x
∣
≤
a
,
|x|\leq a,
∣
x
∣
≤
a
,
find the maximum and minimum value of
∫
x
−
a
x
+
a
4
a
2
−
t
2
d
t
.
\int_{x-a}^{x+a}\sqrt{4a^{2}-t^{2}}\ dt.
∫
x
−
a
x
+
a
4
a
2
−
t
2
d
t
.
113
1
Hide problems
Today's calculation of Integral 113
Evaluate
∫
π
4
π
3
sin
x
+
cos
x
+
3
(
sin
x
−
cos
x
)
cos
2
x
sin
2
x
d
x
.
\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sqrt{\sin x}+\sqrt{\cos x}+3(\sqrt{\sin x}-\sqrt{\cos x})\cos 2x}{\sqrt{\sin 2x}}\ dx.
∫
4
π
3
π
s
i
n
2
x
s
i
n
x
+
c
o
s
x
+
3
(
s
i
n
x
−
c
o
s
x
)
c
o
s
2
x
d
x
.
112
1
Hide problems
Today's calculation of Integral 112
Evaluate
∫
0
π
2
d
θ
(
1
−
e
2
sin
2
θ
)
3
(
e
<
1
is a constant number
)
.
\int_0^{\frac{\pi}{2}} \frac{d\theta }{(1-e^2\sin ^ 2 \theta)^3}\ (e<1\ \text{is a constant number}).
∫
0
2
π
(
1
−
e
2
s
i
n
2
θ
)
3
d
θ
(
e
<
1
is a constant number
)
.
111
1
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Today's calculation of Integral 111
Evaluate
∫
−
π
2
π
2
e
−
n
x
cos
m
x
d
x
(
m
,
n
=
0
,
1
,
2
,
⋯
)
.
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} e^{-nx}\cos ^ m x\ dx\ (m,n=0,1,2,\cdots).
∫
−
2
π
2
π
e
−
n
x
cos
m
x
d
x
(
m
,
n
=
0
,
1
,
2
,
⋯
)
.
110
1
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Today's calculation of integral 110
Prove the following inequality.
1
≤
∫
0
π
2
1
−
sin
3
x
d
x
≤
1
2
{
2
+
ln
(
1
+
2
)
}
1\leq \int_0^{\frac{\pi}{2}} \sqrt{1-\sin ^ 3 x} \ dx\leq \frac{1}{2}\{\sqrt{2}+\ln (1+\sqrt{2}\ ) \}
1
≤
∫
0
2
π
1
−
sin
3
x
d
x
≤
2
1
{
2
+
ln
(
1
+
2
)}
109
1
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Today's calculation of integral 109
Let
I
n
=
∫
2006
2006
+
1
n
x
cos
2
(
x
−
2006
)
d
x
(
n
=
1
,
2
,
⋯
)
.
I_n=\int_{2006}^{2006+\frac{1}{n}} x\cos ^ 2 (x-2006)\ dx\ (n=1,2,\cdots).
I
n
=
∫
2006
2006
+
n
1
x
cos
2
(
x
−
2006
)
d
x
(
n
=
1
,
2
,
⋯
)
.
Find
lim
n
→
∞
n
I
n
.
\lim_{n\to\infty} nI_n.
lim
n
→
∞
n
I
n
.
108
1
Hide problems
Today's calculation of integral 108
For
x
≥
0
,
x\geq 0,
x
≥
0
,
find the minimum value of
x
x
x
for which
∫
0
x
2
t
(
2
t
−
3
)
(
x
−
t
)
d
t
\int_0^x 2^t(2^t-3)(x-t)\ dt
∫
0
x
2
t
(
2
t
−
3
)
(
x
−
t
)
d
t
is minimized.
107
1
Hide problems
Today's calculation of integral 107
Evaluate
∫
−
1
1
1
−
x
2
a
−
x
d
x
(
a
>
1
)
\int_{-1}^1 \frac{\sqrt{1-x^2}}{a-x}\ dx\ (a>1)
∫
−
1
1
a
−
x
1
−
x
2
d
x
(
a
>
1
)
106
1
Hide problems
Today's calculation of integral 106
Evaluate
∫
0
1
1
−
x
2
1
+
x
2
d
x
1
+
x
4
\int_0^1 \frac{1-x^2}{1+x^2}\frac{dx}{\sqrt{1+x^4}}
∫
0
1
1
+
x
2
1
−
x
2
1
+
x
4
d
x
105
1
Hide problems
Today's caluculation of integral 105
Let
a
,
b
a,b
a
,
b
be constant numbers such that
0
<
a
<
b
.
0<a<b.
0
<
a
<
b
.
If a function
f
(
x
)
f(x)
f
(
x
)
always satisfies
f
′
(
x
)
>
0
f'(x) >0
f
′
(
x
)
>
0
at
a
<
x
<
b
,
a<x<b,
a
<
x
<
b
,
for
a
<
t
<
b
a<t<b
a
<
t
<
b
find the value of
t
t
t
for which the following the integral is minimized.
∫
a
b
∣
f
(
x
)
−
f
(
t
)
∣
x
d
x
.
\int_a^b |f(x)-f(t)|x\ dx.
∫
a
b
∣
f
(
x
)
−
f
(
t
)
∣
x
d
x
.
104
1
Hide problems
Today's calculation of integral 104
For
0
<
x
<
1
,
0<x<1,
0
<
x
<
1
,
let
f
(
x
)
=
∫
0
x
d
t
1
−
t
2
d
t
f(x)=\int_0^x \frac{dt}{\sqrt{1-t^2}}\ dt
f
(
x
)
=
∫
0
x
1
−
t
2
d
t
d
t
(1) Find
d
d
x
f
(
1
−
x
2
)
\frac{d}{dx} f(\sqrt{1-x^2})
d
x
d
f
(
1
−
x
2
)
(2) Find
f
(
1
2
)
f\left(\frac{1}{\sqrt{2}}\right)
f
(
2
1
)
(3) Prove that
f
(
x
)
+
f
(
1
−
x
2
)
=
π
2
f(x)+f(\sqrt{1-x^2})=\frac{\pi}{2}
f
(
x
)
+
f
(
1
−
x
2
)
=
2
π
103
1
Hide problems
Today's calculation of integral 103
For
0
<
a
<
1
,
0<a<1,
0
<
a
<
1
,
let
f
(
x
)
=
a
−
x
1
−
a
x
(
−
1
<
x
<
1
)
.
f(x)=\frac{a-x}{1-ax}\ (-1<x<1).
f
(
x
)
=
1
−
a
x
a
−
x
(
−
1
<
x
<
1
)
.
Evaluate
∫
0
a
1
−
{
f
(
x
)
}
6
1
−
x
2
d
x
.
\int_0^a \frac{1-\{f(x)\}^6}{1-x^2}\ dx.
∫
0
a
1
−
x
2
1
−
{
f
(
x
)
}
6
d
x
.
102
1
Hide problems
Today's calculation of integral 102
Let
a
,
b
a,b
a
,
b
be costant numbers such that
a
2
≥
b
.
a^2\geq b.
a
2
≥
b
.
Find the following indefinite integrals. (1)
I
=
∫
d
x
x
2
+
2
a
x
+
b
I=\int \frac{dx}{x^2+2ax+b}
I
=
∫
x
2
+
2
a
x
+
b
d
x
(2)
J
=
∫
d
x
(
x
2
+
2
a
x
+
b
)
2
J=\int \frac{dx}{(x^2+2ax+b)^2}
J
=
∫
(
x
2
+
2
a
x
+
b
)
2
d
x
101
1
Hide problems
Today's calculation of integral 101
Thank you very much, vidyamanohar. I will continue to post problems. For
n
>
2
,
n>2,
n
>
2
,
prove the following inequality.
1
2
<
∫
0
1
2
1
1
−
x
n
d
x
<
π
6
.
\frac{1}{2}<\int_0^{\frac{1}{2}} \frac{1}{\sqrt{1-x^n}} dx<\frac{\pi}{6}.
2
1
<
∫
0
2
1
1
−
x
n
1
d
x
<
6
π
.
100
1
Hide problems
Today's calculation of integral 100
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive numbers such that
a
b
c
=
1
16
.
abc=\frac{1}{16}.
ab
c
=
16
1
.
Prove the following inequality.
∫
0
∞
x
2
(
x
2
+
a
2
)
(
x
2
+
b
2
)
(
x
2
+
c
2
)
d
x
≤
π
.
\int_0^{\infty} \frac{x^2}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}\ dx\leq \pi.
∫
0
∞
(
x
2
+
a
2
)
(
x
2
+
b
2
)
(
x
2
+
c
2
)
x
2
d
x
≤
π
.
99
1
Hide problems
Today's calculation of integral 99
Let
θ
\theta
θ
be a constant number such that
0
≤
θ
≤
π
.
0\leq \theta \leq \pi.
0
≤
θ
≤
π
.
Evaluate
∫
0
2
π
sin
8
x
∣
sin
(
x
−
θ
)
∣
d
x
.
\int_0^{2\pi} \sin 8x|\sin (x-\theta)|\ dx.
∫
0
2
π
sin
8
x
∣
sin
(
x
−
θ
)
∣
d
x
.
98
1
Hide problems
Today's calculation of integral 98
Let
I
n
=
∫
1
1
+
1
n
{
[
(
x
+
1
)
ln
x
+
1
]
e
x
(
e
x
ln
x
+
1
)
+
n
}
d
x
(
n
=
1
,
2
,
⋯
)
.
{{ \ I_n=\int_1^{1+\frac{1}{n}}\{[(x+1)\ln x+1]}e^{x(e^{x}\ln x+1)}}+n\}dx \ (n=1,2,\cdots).
I
n
=
∫
1
1
+
n
1
{[(
x
+
1
)
ln
x
+
1
]
e
x
(
e
x
l
n
x
+
1
)
+
n
}
d
x
(
n
=
1
,
2
,
⋯
)
.
Evaluate
lim
n
→
∞
I
n
n
.
{\lim_{n\to\infty}I_n^{n}}.
lim
n
→
∞
I
n
n
.
97
1
Hide problems
Today's calculation of integral 97
Answer the following questions. (1) Evaluate
∫
e
e
e
ln
(
ln
x
)
x
ln
x
d
x
.
\int_e^{e^e} \frac{\ln (\ln x)}{x\ln x} dx.
∫
e
e
e
x
l
n
x
l
n
(
l
n
x
)
d
x
.
(2) Let
α
,
β
\alpha,\beta
α
,
β
be real numbers.Find the values of
α
,
β
\alpha,\beta
α
,
β
for which the following equality holds for any real numbers
p
,
q
.
p,q.
p
,
q
.
∫
−
π
2
π
2
(
p
cos
x
+
q
sin
x
)
(
x
2
+
α
x
+
β
)
d
x
=
0.
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(p\cos x+q\sin x)(x^2+\alpha x+\beta) dx=0.
∫
−
2
π
2
π
(
p
cos
x
+
q
sin
x
)
(
x
2
+
αx
+
β
)
d
x
=
0.
96
1
Hide problems
Today's calculation of integral 96
For
a
≥
0
,
a\geq 0,
a
≥
0
,
find the minimum value of
∫
−
2
1
∣
x
2
+
2
a
x
∣
d
x
.
\int_{-2}^1 |x^2+2ax| dx.
∫
−
2
1
∣
x
2
+
2
a
x
∣
d
x
.
95
1
Hide problems
Today's calculation of integral 95
Evaluate
∫
π
4
π
3
2
sin
3
x
cos
5
x
d
x
.
\int_\frac{\pi}{4}^\frac{\pi}{3} \frac{2\sin ^ 3 x}{\cos ^ 5 x} dx.
∫
4
π
3
π
cos
5
x
2
sin
3
x
d
x
.
94
1
Hide problems
Today's calculation of integral 94
Let
a
a
a
be real numbers.Find the following limit value.
lim
T
→
∞
1
T
∫
0
T
(
sin
x
+
sin
a
x
)
2
d
x
.
\lim_{T\rightarrow \infty} \frac{1}{T} \int_0^T (\sin x+\sin ax)^2 dx.
T
→
∞
lim
T
1
∫
0
T
(
sin
x
+
sin
a
x
)
2
d
x
.
93
1
Hide problems
Today's calculation of integral 93
Evaluate
1
∫
0
π
2
cos
2005
x
sin
2007
x
d
x
.
\frac{1}{\displaystyle \int_0^{\frac{\pi}{2}} \cos ^{2005}x\ \sin {2007x}\ dx}.
∫
0
2
π
cos
2005
x
sin
2007
x
d
x
1
.
92
1
Hide problems
Today's calculation of integral 92
Evaluate
lim
n
→
∞
n
2
∫
−
1
n
1
n
(
2005
sin
x
+
2006
cos
x
)
∣
x
∣
d
x
.
\lim_{n\to\infty} n^2\int_{-\frac{1}{n}}^{\frac{1}{n}} (2005\sin x+2006\cos x)|x|dx.
lim
n
→
∞
n
2
∫
−
n
1
n
1
(
2005
sin
x
+
2006
cos
x
)
∣
x
∣
d
x
.