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Today's Calculation Of Integral
2007 Today's Calculation Of Integral
2007 Today's Calculation Of Integral
Part of
Today's Calculation Of Integral
Subcontests
(89)
245
1
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Today's calculation of Integral 245
A sextic funtion y \equal{} ax^6 \plus{} bx^5 \plus{} cx^4 \plus{} dx^3 \plus{} ex^2 \plus{} fx \plus{} g\ (a\neq 0) touches the line y \equal{} px \plus{} q at x \equal{} \alpha ,\ \beta ,\ \gamma \ (\alpha < \beta < \gamma ). Find the area of the region bounded by these graphs in terms of
a
,
α
,
β
,
γ
.
a,\ \alpha ,\ \beta ,\gamma .
a
,
α
,
β
,
γ
.
created by kunny
236
1
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Today's calculation of Integral 236
Let
a
a
a
be a positive constant. Evaluate the following definite integrals
A
,
B
A,\ B
A
,
B
.
A
=
∫
0
π
e
−
a
x
sin
2
x
d
x
,
B
=
∫
0
π
e
−
a
x
cos
2
x
d
x
A=\int_0^{\pi} e^{-ax}\sin ^ 2 x\ dx,\ B=\int_0^{\pi} e^{-ax}\cos ^ 2 x\ dx
A
=
∫
0
π
e
−
a
x
sin
2
x
d
x
,
B
=
∫
0
π
e
−
a
x
cos
2
x
d
x
.1998 Shinsyu University entrance exam/Textile Science
256
1
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Today's calculation of Integral 256
Find the value of
a
a
a
for which \int_0^{\pi} \{ax(\pi ^ 2 \minus{} x^2) \minus{} \sin x\}^2dx is minimized.
255
1
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Today's calculation of Integral 255
Find the value of
a
a
a
for which the area of the figure surrounded by y \equal{} e^{ \minus{} x} and y \equal{} ax \plus{} 3\ (a < 0) is minimized.
254
1
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Today's calculation of Integral 254
Evaluate \int_e^{e^2} \frac {(\ln x)^7\minus{}7!}{(\ln x)^8}\ dx. Sorry, I have deleted my first post because that was wrong. kunny
253
1
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Today's calculation of Integral 253
Evaluate \int_0^1 (1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{n \minus{} 1})\{1 \plus{} 3x \plus{} 5x^2 \plus{} \cdots \plus{} (2n \minus{} 3)x^{n \minus{} 2} \plus{} (2n \minus{} 1)x^{n \minus{} 1}\}\ dx.
252
1
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Today's calculation of Integral 252
Compare \displaystyle f(\theta) \equal{} \int_0^1 (x \plus{} \sin \theta)^2\ dx and \ g(\theta) \equal{} \int_0^1 (x \plus{} \cos \theta)^2\ dx for
0
≦
θ
≦
2
π
.
0\leqq \theta \leqq 2\pi .
0
≦
θ
≦
2
π
.
251
1
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Today's calculation of Integral 251
Evaluate \int_0^{n\pi} e^x\sin ^ 4 x\ dx\ (n\equal{}1,\ 2,\ \cdots).
250
1
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Today's calculation of Integral 250
For a positive constant number
p
p
p
, find \lim_{n\to\infty} \frac {1}{n^{p \plus{} 1}}\sum_{k \equal{} 0}^{n \minus{} 1} \int_{2k\pi}^{(2k \plus{} 1)\pi} x^p\sin ^ 3 x\cos ^ 2x\ dx.
249
1
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Today's calculation of Integral 249
Determine the sign of \int_{\frac{1}{2}}^2 \frac{\ln t}{1\plus{}t^n}\ dt\ (n\equal{}1, 2, \cdots).
248
1
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Today's calculation of Integral 248
Evaluate
∫
π
4
3
4
π
cos
1
sin
(
1
sin
x
)
⋅
cos
(
1
sin
x
)
⋅
cos
x
sin
2
x
⋅
sin
2
(
1
sin
x
)
d
x
\int_{\frac {\pi}{4}}^{\frac {3}{4}\pi } \cos \frac {1}{\sin \left(\frac {1}{\sin x}\right)}\cdot \cos \left(\frac {1}{\sin x}\right)\cdot \frac {\cos x}{\sin ^ 2 x\cdot \sin ^ 2 \left(\frac {1}{\sin x }\right)}\ dx
∫
4
π
4
3
π
cos
s
i
n
(
s
i
n
x
1
)
1
⋅
cos
(
s
i
n
x
1
)
⋅
s
i
n
2
x
⋅
s
i
n
2
(
s
i
n
x
1
)
c
o
s
x
d
x
Last Edited, Sorry kunny
247
1
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Today's calculation of Integral 247
Evaluate \int_{\frac{\pi}{8}}^{\frac{3}{8}\pi} \frac{11\plus{}4\cos 2x \plus{}\cos 4x}{1\minus{}\cos 4x}\ dx.
246
1
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Today's calculation of Integral 246
An eighth degree polynomial funtion y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0) touches the line y \equal{} px \plus{} q at x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \ (\alpha < \beta < \gamma <\delta). Find the area of the region bounded by these graphs in terms of
a
,
α
,
β
,
γ
,
δ
.
a,\ \alpha ,\ \beta ,\gamma ,\ \delta .
a
,
α
,
β
,
γ
,
δ
.
244
1
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Today's calculation of Integral 244
A quartic funtion y \equal{} ax^4 \plus{} bx^3 \plus{} cx^2 \plus{} dx\plus{}e\ (a\neq 0) touches the line y \equal{} px \plus{} q at x \equal{} \alpha ,\ \beta \ (\alpha < \beta ). Find the area of the region bounded by these graphs in terms of
a
,
α
,
β
a,\ \alpha ,\ \beta
a
,
α
,
β
.
243
1
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Today's calculation of Integral 243
A cubic funtion y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0) intersects with the line y \equal{} px \plus{} q at x \equal{} \alpha ,\ \beta ,\ \gamma\ (\alpha < \beta < \gamma). Find the area of the region bounded by these graphs in terms of
a
,
α
,
β
,
γ
a,\ \alpha ,\ \beta ,\ \gamma
a
,
α
,
β
,
γ
.
242
1
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Today's calculation of Integral 242
A cubic function y \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d\ (a\neq 0) touches a line y \equal{} px \plus{} q at x \equal{} \alpha and intersects x \equal{} \beta \ (\alpha \neq \beta). Find the area of the region bounded by these graphs in terms of
a
,
α
,
β
a,\ \alpha ,\ \beta
a
,
α
,
β
.
241
1
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Today's calculation of Integral 241
1.Let x \equal{} \alpha ,\ \beta \ (\alpha < \beta ) are
x
x
x
coordinates of the intersection points of a parabola y \equal{} ax^2 \plus{} bx \plus{} c\ (a\neq 0) and the line y \equal{} ux \plus{} v. Prove that the area of the region bounded by these graphs is \boxed{\frac {|a|}{6}(\beta \minus{} \alpha )^3}. 2. Let x \equal{} \alpha ,\ \beta \ (\alpha < \beta ) are
x
x
x
coordinates of the intersection points of parabolas y \equal{} ax^2 \plus{} bx \plus{} c and y \equal{} px^2 \plus{} qx \plus{} r\ (ap\neq 0). Prove that the area of the region bounded by these graphs is \boxed{\frac {|a \minus{} p|}{6}(\beta \minus{} \alpha )^3}.
240
1
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Today's calculation of Integral 240
2 curves y \equal{} x^3 \minus{} x and y \equal{} x^2 \minus{} a pass through the point
P
P
P
and have a common tangent line at
P
P
P
. Find the area of the region bounded by these curves.
239
1
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Today's calculation of Integral 239
Evaluate
∫
0
π
sin
(
π
cos
x
)
d
x
.
\int_0^{\pi} \sin (\pi \cos x)\ dx.
∫
0
π
sin
(
π
cos
x
)
d
x
.
238
1
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Today's calculation of Integral 238
Find \lim_{a\to\infty} \frac {1}{a^2}\int_0^a \log (1 \plus{} e^x)\ dx.
237
1
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Today's calculation of Integral 237
Calculate \int \frac {dx}{x^{2008}(1 \minus{} x)}
235
1
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Today's calculation of Integral 235
Show that a function f(x)\equal{}\int_{\minus{}1}^1 (1\minus{}|\ t\ |)\cos (xt)\ dt is continuous at x\equal{}0.
234
1
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Today's calculation of Integral 234
For
x
≥
0
,
x\geq 0,
x
≥
0
,
define a function f(x)\equal{}\sin \left(\frac{n\pi}{4}\right)\sin x\ (n\pi \leq x<(n\plus{}1)\pi )\ (n\equal{}0,\ 1,\ 2,\ \cdots). Evaluate
∫
0
100
π
f
(
x
)
d
x
.
\int_0^{100\pi } f(x)\ dx.
∫
0
100
π
f
(
x
)
d
x
.
233
1
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Today's calculation of Integral 233
Find the minimum value of the following definite integral. \int_0^{\pi} (a\sin x \plus{} b\sin 3x \minus{} 1)^2\ dx.
232
1
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Today's calculation of Integral 232
For f(x)\equal{}1\minus{}\sin x, let g(x)\equal{}\int_0^x (x\minus{}t)f(t)\ dt. Show that g(x\plus{}y)\plus{}g(x\minus{}y)\geq 2g(x) for any real numbers
x
,
y
.
x,\ y.
x
,
y
.
231
1
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Today's calculation of Integral 231
Evaluate
∫
0
π
3
1
cos
7
x
d
x
\int_0^{\frac{\pi}{3}} \frac{1}{\cos ^ 7 x}\ dx
∫
0
3
π
c
o
s
7
x
1
d
x
.
230
1
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Today's calculation of Integral 230
Prove that \frac {( \minus{} 1)^n}{n!}\int_1^2 (\ln x)^n\ dx \equal{} 2\sum_{k \equal{} 1}^n \frac {( \minus{} \ln 2)^k}{k!} \plus{} 1.
229
1
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Today's calculation of Integral 229
Find \lim_{a\rightarrow \plus{} \infty} \frac {\int_0^a \sin ^ 4 x\ dx}{a}.
228
1
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Today's calculation of Integral 228
Let x_n \equal{} \int_0^{\frac {\pi}{2}} \sin ^ n \theta \ d\theta \ (n \equal{} 0,\ 1,\ 2,\ \cdots). (1) Show that x_n \equal{} \frac {n \minus{} 1}{n}x_{n \minus{} 2}. (2) Find the value of nx_nx_{n \minus{} 1}. (3) Show that a sequence
{
x
n
}
\{x_n\}
{
x
n
}
is monotone decreasing. (4) Find
lim
n
→
∞
n
x
n
2
\lim_{n\to\infty} nx_n^2
lim
n
→
∞
n
x
n
2
.
227
1
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Today's calculation of Integral 227
Evaluate
1
∫
0
π
2
cos
2006
x
⋅
sin
2008
x
d
x
\frac{1}{\displaystyle \int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\ dx}
∫
0
2
π
cos
2006
x
⋅
sin
2008
x
d
x
1
226
1
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Today's calculation of Integral 226
Evaluate \int_0^{\frac {\pi}{2}} \frac {x^2}{(\cos x \plus{} x\sin x)^2}\ dx Virgil Nicula have already posted the integral :oops:
225
1
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Today's calculation of Integral 225
2 Points
P
(
a
,
1
a
)
,
Q
(
2
a
,
1
2
a
)
(
a
>
0
)
P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)
P
(
a
,
a
1
)
,
Q
(
2
a
,
2
a
1
)
(
a
>
0
)
are on the curve C: y \equal{}\frac{1}{x}. Let
l
,
m
l,\ m
l
,
m
be the tangent lines at
P
,
Q
P,\ Q
P
,
Q
respectively. Find the area of the figure surrounded by
l
,
m
l,\ m
l
,
m
and
C
C
C
.
224
1
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Today's calculation of Integral 224
Let f(x)\equal{}x^{2}\plus{}|x|. Prove that \int_{0}^{\pi}f(\cos x)\ dx\equal{}2\int_{0}^{\frac{\pi}{2}}f(\sin x)\ dx.
223
1
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Today's calculation of Integral 223
Evaluate \int_{0}^{\pi}\sqrt{(\cos x\plus{}\cos 2x\plus{}\cos 3x)^{2}\plus{}(\sin x\plus{}\sin 2x\plus{}\sin 3x)^{2}}\ dx.
222
1
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Today's calculation of Integral 222
Find \lim_{a\rightarrow\infty}\int_{a}^{a\plus{}1}\frac{x}{x\plus{}\ln x}\ dx.
221
1
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Today's calculation of Integral 221
Evaluate \int_{2}^{6}\ln\frac{\minus{}1\plus{}\sqrt{1\plus{}4x}}{2}\ dx.
220
1
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Today's calculation of Integral 220
Prove that \frac{\pi}{2}\minus{}1<\int_{0}^{1}e^{\minus{}2x^{2}}\ dx.
219
1
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Today's calculation of Integral 219
Let f(x)\equal{}\left(1\plus{}\frac{1}{x}\right)^{x}\ (x>0). Find
lim
n
→
∞
{
f
(
1
n
)
f
(
2
n
)
f
(
3
n
)
⋯
⋯
f
(
n
n
)
}
1
n
\lim_{n\to\infty}\left\{f\left(\frac{1}{n}\right)f\left(\frac{2}{n}\right)f\left(\frac{3}{n}\right)\cdots\cdots f\left(\frac{n}{n}\right)\right\}^{\frac{1}{n}}
lim
n
→
∞
{
f
(
n
1
)
f
(
n
2
)
f
(
n
3
)
⋯⋯
f
(
n
n
)
}
n
1
.
218
1
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Today's calculation of Integral 218
For any quadratic functions
f
(
x
)
f(x)
f
(
x
)
such that f'(2)\equal{}1, evaluate \int_{2\minus{}\pi}^{2\plus{}\pi}f(x)\sin\left(\frac{x}{2}\minus{}1\right) dx.
217
1
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Today's calculation of Integral 217
Evaluate \int_{0}^{1}e^{\sqrt{e^{x}}}\ dx\plus{}2\int_{e}^{e^{\sqrt{e}}}\ln (\ln x)\ dx.
216
1
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Today's calculation of Integral 216
Let
a
n
a_{n}
a
n
is a positive number such that \int_{0}^{a_{n}}\frac{e^{x}\minus{}1}{1\plus{}e^{x}}\ dx \equal{}\ln n. Find \lim_{n\to\infty}(a_{n}\minus{}\ln n).
215
1
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Today's calculation of Integral 215
For
a
∈
R
a\in\mathbb{R}
a
∈
R
, let
M
(
a
)
M(a)
M
(
a
)
be the maximum value of the function f(x)\equal{}\int_{0}^{\pi}\sin (x\minus{}t)\sin (2t\minus{}a)\ dt. Evaluate
∫
0
π
2
M
(
a
)
sin
(
2
a
)
d
a
\int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da
∫
0
2
π
M
(
a
)
sin
(
2
a
)
d
a
.
214
1
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Today's calculation of Integral 214
Find the area of the region surrounded by the two curves
y
=
x
,
x
+
y
=
1
y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1
y
=
x
,
x
+
y
=
1
and the
x
x
x
axis.
213
1
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Today's calculation of Integral 213
Find the minimum value of
f
(
a
)
=
∫
0
1
x
∣
x
−
a
∣
d
x
f(a)=\int_{0}^{1}x|x-a|\ dx
f
(
a
)
=
∫
0
1
x
∣
x
−
a
∣
d
x
.
212
1
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Today's calculation of Integral 212
For integers
k
(
0
≤
k
≤
5
)
k\ (0\leq k\leq 5)
k
(
0
≤
k
≤
5
)
, positive numbers
m
,
n
m,\ n
m
,
n
and real numbers
a
,
b
a,\ b
a
,
b
, let
f
(
k
)
=
∫
−
π
π
(
sin
k
x
−
a
sin
m
x
−
b
sin
n
x
)
2
d
x
f(k)=\int_{-\pi}^{\pi}(\sin kx-a\sin mx-b\sin nx)^{2}\ dx
f
(
k
)
=
∫
−
π
π
(
sin
k
x
−
a
sin
m
x
−
b
sin
n
x
)
2
d
x
,
p
(
k
)
=
5
!
k
!
(
5
−
k
)
!
(
1
2
)
5
,
E
=
∑
k
=
0
5
p
(
k
)
f
(
k
)
p(k)=\frac{5!}{k!(5-k)!}\left(\frac{1}{2}\right)^{5}, \ E=\sum_{k=0}^{5}p(k)f(k)
p
(
k
)
=
k
!
(
5
−
k
)!
5
!
(
2
1
)
5
,
E
=
∑
k
=
0
5
p
(
k
)
f
(
k
)
. Find the values of
m
,
n
,
a
,
b
m,\ n,\ a,\ b
m
,
n
,
a
,
b
for which
E
E
E
is minimized.
211
1
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Today's calculation of Integral 211
When the parabola which has the axis parallel to
y
y
y
-axis and passes through the origin touch to the rectangular hyperbola
x
y
=
1
xy=1
x
y
=
1
in the first quadrant moves, prove that the area of the figure sorrounded by the parabola and the
x
x
x
-axis is constant.
210
1
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Today's calculation of Integral 210
Evaluate
∫
1
π
(
x
3
ln
x
−
6
x
)
sin
x
d
x
\int_{1}^{\pi}\left(x^{3}\ln x-\frac{6}{x}\right)\sin x\ dx
∫
1
π
(
x
3
ln
x
−
x
6
)
sin
x
d
x
.
209
1
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Today's calculation of Integral 209
Let
m
,
n
m,\ n
m
,
n
be the given distinct positive integers. Answer the following questions. (1) Find the real number
α
(
∣
α
∣
<
1
)
\alpha \ (|\alpha |<1)
α
(
∣
α
∣
<
1
)
such that
∫
−
π
π
sin
(
m
+
α
)
x
sin
(
n
+
α
)
x
d
x
=
0
\int_{-\pi}^{\pi}\sin (m+\alpha )x\ \sin (n+\alpha )x\ dx=0
∫
−
π
π
sin
(
m
+
α
)
x
sin
(
n
+
α
)
x
d
x
=
0
. (2) Find the real number
β
\beta
β
satifying the sytem of equation
∫
−
π
π
sin
2
(
m
+
β
)
x
d
x
=
π
+
2
4
m
−
1
\int_{-\pi}^{\pi}\sin^{2}(m+\beta )x\ dx=\pi+\frac{2}{4m-1}
∫
−
π
π
sin
2
(
m
+
β
)
x
d
x
=
π
+
4
m
−
1
2
,
∫
−
π
π
sin
2
(
n
+
β
)
x
d
x
=
π
+
2
4
n
−
1
\int_{-\pi}^{\pi}\sin^{2}(n+\beta )x\ dx=\pi+\frac{2}{4n-1}
∫
−
π
π
sin
2
(
n
+
β
)
x
d
x
=
π
+
4
n
−
1
2
.
208
1
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Today's caluculation of Integral 208
Find the values of real numbers
a
,
b
a,\ b
a
,
b
for which the function
f
(
x
)
=
a
∣
cos
x
∣
+
b
∣
sin
x
∣
f(x)=a|\cos x|+b|\sin x|
f
(
x
)
=
a
∣
cos
x
∣
+
b
∣
sin
x
∣
has local minimum at
x
=
−
π
3
x=-\frac{\pi}{3}
x
=
−
3
π
and satisfies
∫
−
π
2
π
2
{
f
(
x
)
}
2
d
x
=
2
\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\{f(x)\}^{2}dx=2
∫
−
2
π
2
π
{
f
(
x
)
}
2
d
x
=
2
.
207
1
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Today's calculation of Integral 207
Evaluate the following definite integral.
∫
e
e
e
e
+
1
{
1
ln
x
⋅
ln
(
ln
x
)
+
ln
(
ln
(
ln
x
)
)
}
d
x
\int_{e^{e}}^{e^{e+1}}\left\{\frac{1}{\ln x \cdot\ln (\ln x)}+\ln (\ln (\ln x))\right\}dx
∫
e
e
e
e
+
1
{
ln
x
⋅
ln
(
ln
x
)
1
+
ln
(
ln
(
ln
x
))
}
d
x
206
1
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Today's calculation of Integral 206
Calculate
∫
x
3
(
x
−
1
)
3
(
x
−
2
)
d
x
\int \frac{x^{3}}{(x-1)^{3}(x-2)}\ dx
∫
(
x
−
1
)
3
(
x
−
2
)
x
3
d
x
205
1
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Today's calculation of Integral 205
Evaluate the following definite integral.
∫
e
2
e
3
ln
x
⋅
ln
(
x
ln
x
)
⋅
ln
{
x
ln
(
x
ln
x
)
}
+
ln
x
+
1
ln
x
⋅
ln
(
x
ln
x
)
d
x
\int_{e^{2}}^{e^{3}}\frac{\ln x\cdot \ln (x\ln x)\cdot \ln \{x\ln (x\ln x)\}+\ln x+1}{\ln x\cdot \ln (x\ln x)}\ dx
∫
e
2
e
3
ln
x
⋅
ln
(
x
ln
x
)
ln
x
⋅
ln
(
x
ln
x
)
⋅
ln
{
x
ln
(
x
ln
x
)}
+
ln
x
+
1
d
x
204
1
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Today's calculation of Integral 204
Evaluate
∫
0
1
x
d
x
(
x
2
+
x
+
1
)
3
2
\int_{0}^{1}\frac{x\ dx}{(x^{2}+x+1)^{\frac{3}{2}}}
∫
0
1
(
x
2
+
x
+
1
)
2
3
x
d
x
203
1
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Today's calculation of Integral 203
Let
α
,
β
\alpha ,\ \beta
α
,
β
be the distinct positive roots of the equation of
2
x
=
tan
x
2x=\tan x
2
x
=
tan
x
. Evaluate the following definite integral.
∫
0
1
sin
α
x
sin
β
x
d
x
\int_{0}^{1}\sin \alpha x\sin \beta x\ dx
∫
0
1
sin
αx
sin
β
x
d
x
202
1
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Today's calculation of Integral 202
Let
a
,
b
a,\ b
a
,
b
are real numbers such that
a
+
b
=
1
a+b=1
a
+
b
=
1
. Find the minimum value of the following integral.
∫
0
π
(
a
sin
x
+
b
sin
2
x
)
2
d
x
\int_{0}^{\pi}(a\sin x+b\sin 2x)^{2}\ dx
∫
0
π
(
a
sin
x
+
b
sin
2
x
)
2
d
x
201
1
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Today's calculation of Integral 201
Evaluate the following definite integral.
∫
−
1
1
e
2
x
+
1
−
(
x
+
1
)
(
e
x
+
e
−
x
)
x
(
e
x
−
1
)
d
x
\int_{-1}^{1}\frac{e^{2x}+1-(x+1)(e^{x}+e^{-x})}{x(e^{x}-1)}dx
∫
−
1
1
x
(
e
x
−
1
)
e
2
x
+
1
−
(
x
+
1
)
(
e
x
+
e
−
x
)
d
x
200
1
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Today's calculation of Integral 200
Evaluate the following definite integral.
∫
0
π
cos
n
x
2
−
cos
x
d
x
(
n
=
0
,
1
,
2
,
⋯
)
\int_{0}^{\pi}\frac{\cos nx}{2-\cos x}dx\ (n=0,\ 1,\ 2,\ \cdots)
∫
0
π
2
−
cos
x
cos
n
x
d
x
(
n
=
0
,
1
,
2
,
⋯
)
199
1
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Today's calculation of Integral 199
Let
m
,
n
m,\ n
m
,
n
be non negative integers. Calculate
∑
k
=
0
n
(
−
1
)
k
n
+
m
+
1
k
+
m
+
1
n
C
k
.
\sum_{k=0}^{n}(-1)^{k}\frac{n+m+1}{k+m+1}\ nC_{k}.
k
=
0
∑
n
(
−
1
)
k
k
+
m
+
1
n
+
m
+
1
n
C
k
.
where
i
C
j
_{i}C_{j}
i
C
j
is a binomial coefficient which means
i
⋅
(
i
−
1
)
⋯
(
i
−
j
+
1
)
j
⋅
(
j
−
1
)
⋯
2
⋅
1
\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}
j
⋅
(
j
−
1
)
⋯
2
⋅
1
i
⋅
(
i
−
1
)
⋯
(
i
−
j
+
1
)
.
198
1
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Today's calculation of Integral 198
Compare the values of the following definite integrals.
∫
0
∞
ln
(
x
+
1
x
)
d
x
1
+
x
2
,
∫
0
π
2
(
θ
sin
θ
)
2
d
θ
\int_{0}^{\infty}\ln \left(x+\frac{1}{x}\right)\frac{dx}{1+x^{2}},\ \ \int_{0}^{\frac{\pi}{2}}\left(\frac{\theta}{\sin \theta}\right)^{2}d\theta
∫
0
∞
ln
(
x
+
x
1
)
1
+
x
2
d
x
,
∫
0
2
π
(
sin
θ
θ
)
2
d
θ
197
1
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Today's calculation of Integral 197
Let
∣
a
∣
<
π
2
.
|a|<\frac{\pi}{2}.
∣
a
∣
<
2
π
.
Evaluate the following definite integral.
∫
0
π
2
d
x
{
sin
(
a
+
x
)
+
cos
x
}
2
\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}
∫
0
2
π
{
sin
(
a
+
x
)
+
cos
x
}
2
d
x
196
1
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Today's calculation of Integral 196
Calculate
∫
0
π
e
−
x
sin
n
x
d
x
∫
0
π
e
x
sin
n
x
d
x
(
n
=
1
,
2
,
⋯
)
.
\frac{\int_{0}^{\pi}e^{-x}\sin^{n}x\ dx}{\int_{0}^{\pi}e^{x}\sin^{n}x \ dx}\ (n=1,\ 2,\ \cdots).
∫
0
π
e
x
sin
n
x
d
x
∫
0
π
e
−
x
sin
n
x
d
x
(
n
=
1
,
2
,
⋯
)
.
195
1
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Today's calculation of Integral 195
Find continuous functions
x
(
t
)
,
y
(
t
)
x(t),\ y(t)
x
(
t
)
,
y
(
t
)
such that
x
(
t
)
=
1
+
∫
0
t
e
−
2
(
t
−
s
)
x
(
s
)
d
s
\ \ \ \ \ \ \ \ \ x(t)=1+\int_{0}^{t}e^{-2(t-s)}x(s)ds
x
(
t
)
=
1
+
∫
0
t
e
−
2
(
t
−
s
)
x
(
s
)
d
s
y
(
t
)
=
∫
0
t
e
−
2
(
t
−
s
)
{
2
x
(
s
)
+
3
y
(
s
)
}
d
s
\ \ \ \ \ \ \ \ \ y(t)=\int_{0}^{t}e^{-2(t-s)}\{2x(s)+3y(s)\}ds
y
(
t
)
=
∫
0
t
e
−
2
(
t
−
s
)
{
2
x
(
s
)
+
3
y
(
s
)}
d
s
194
1
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Today's calculation of Integral 194
Evaluate
∑
n
=
0
2006
∫
0
1
d
x
2
(
x
+
n
+
1
)
(
x
+
n
)
(
x
+
n
+
1
)
\sum_{n=0}^{2006}\int_{0}^{1}\frac{dx}{2(x+n+1)\sqrt{(x+n)(x+n+1)}}
n
=
0
∑
2006
∫
0
1
2
(
x
+
n
+
1
)
(
x
+
n
)
(
x
+
n
+
1
)
d
x
193
1
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Today's calculation of Integral 193
For
a
>
0
a>0
a
>
0
, let
l
l
l
be the line created by rotating the tangent line to parabola
y
=
x
2
y=x^{2}
y
=
x
2
, which is tangent at point
A
(
a
,
a
2
)
A(a,a^{2})
A
(
a
,
a
2
)
, around
A
A
A
by
−
π
6
-\frac{\pi}{6}
−
6
π
. Let
B
B
B
be the other intersection of
l
l
l
and
y
=
x
2
y=x^{2}
y
=
x
2
. Also, let
C
C
C
be
(
a
,
0
)
(a,0)
(
a
,
0
)
and let
O
O
O
be the origin. (1) Find the equation of
l
l
l
. (2) Let
S
(
a
)
S(a)
S
(
a
)
be the area of the region bounded by
O
C
OC
OC
,
C
A
CA
C
A
and
y
=
x
2
y=x^{2}
y
=
x
2
. Let
T
(
a
)
T(a)
T
(
a
)
be the area of the region bounded by
A
B
AB
A
B
and
y
=
x
2
y=x^{2}
y
=
x
2
. Find
lim
a
→
∞
T
(
a
)
S
(
a
)
\lim_{a \to \infty}\frac{T(a)}{S(a)}
lim
a
→
∞
S
(
a
)
T
(
a
)
.
192
1
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Today's calculation of Integral 192
Let
t
t
t
be positive number. Draw two tangent lines to the palabola
y
=
x
2
y=x^{2}
y
=
x
2
from the point
(
t
,
−
1
)
.
(t,-1).
(
t
,
−
1
)
.
Denote the area of the region bounded by these tangent lines and the parabola by
S
(
t
)
.
S(t).
S
(
t
)
.
Find the minimum value of
S
(
t
)
t
.
\frac{S(t)}{\sqrt{t}}.
t
S
(
t
)
.
191
1
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Today's calculation of Integral 191
(1) For integer
n
=
0
,
1
,
2
,
⋯
n=0,\ 1,\ 2,\ \cdots
n
=
0
,
1
,
2
,
⋯
and positive number
a
n
,
a_{n},
a
n
,
let
f
n
(
x
)
=
a
n
(
x
−
n
)
(
n
+
1
−
x
)
.
f_{n}(x)=a_{n}(x-n)(n+1-x).
f
n
(
x
)
=
a
n
(
x
−
n
)
(
n
+
1
−
x
)
.
Find
a
n
a_{n}
a
n
such that the curve
y
=
f
n
(
x
)
y=f_{n}(x)
y
=
f
n
(
x
)
touches to the curve
y
=
e
−
x
.
y=e^{-x}.
y
=
e
−
x
.
(2) For
f
n
(
x
)
f_{n}(x)
f
n
(
x
)
defined in (1), denote the area of the figure bounded by
y
=
f
0
(
x
)
,
y
=
e
−
x
y=f_{0}(x), y=e^{-x}
y
=
f
0
(
x
)
,
y
=
e
−
x
and the
y
y
y
-axis by
S
0
,
S_{0},
S
0
,
for
n
≥
1
,
n\geq 1,
n
≥
1
,
the area of the figure bounded by
y
=
f
n
−
1
(
x
)
,
y
=
f
n
(
x
)
y=f_{n-1}(x),\ y=f_{n}(x)
y
=
f
n
−
1
(
x
)
,
y
=
f
n
(
x
)
and
y
=
e
−
x
y=e^{-x}
y
=
e
−
x
by
S
n
.
S_{n}.
S
n
.
Find
lim
n
→
∞
(
S
0
+
S
1
+
⋯
+
S
n
)
.
\lim_{n\to\infty}(S_{0}+S_{1}+\cdots+S_{n}).
lim
n
→
∞
(
S
0
+
S
1
+
⋯
+
S
n
)
.
190
1
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Today's calculation of Integral 190
In
x
y
z
xyz
x
yz
space, let
l
l
l
be the segment joining two points
(
1
,
0
,
1
)
(1,\ 0,\ 1)
(
1
,
0
,
1
)
and
(
1
,
0
,
2
)
,
(1,\ 0,\ 2),
(
1
,
0
,
2
)
,
and
A
A
A
be the figure obtained by revolving
l
l
l
around the
z
z
z
axis. Find the volume of the solid obtained by revolving
A
A
A
around the
x
x
x
axis. Note you may not use double integral.
189
1
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Today's calculation of Integral 189
Let
n
n
n
be positive integers. Denote the graph of
y
=
x
y=\sqrt{x}
y
=
x
by
C
,
C,
C
,
and the line passing through two points
(
n
,
n
)
(n,\ \sqrt{n})
(
n
,
n
)
and
(
n
+
1
,
n
+
1
)
(n+1,\ \sqrt{n+1})
(
n
+
1
,
n
+
1
)
by
l
.
l.
l
.
Let
V
V
V
be the volume of the solid obtained by revolving the region bounded by
C
C
C
and
l
l
l
around the
x
x
x
axis.Find the positive numbers
a
,
b
a,\ b
a
,
b
such that
lim
n
→
∞
n
a
V
=
b
.
\lim_{n\to\infty}n^{a}V=b.
lim
n
→
∞
n
a
V
=
b
.
188
1
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Today's calculation of Integral 188
Find the volume of the solid obtained by revolving the region bounded by the graphs of
y
=
x
e
1
−
x
y=xe^{1-x}
y
=
x
e
1
−
x
and
y
=
x
y=x
y
=
x
around the
x
x
x
axis.
187
1
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Today's calculation of Integral 187
For a constant
a
,
a,
a
,
let
f
(
x
)
=
a
x
sin
x
+
x
+
π
2
.
f(x)=ax\sin x+x+\frac{\pi}{2}.
f
(
x
)
=
a
x
sin
x
+
x
+
2
π
.
Find the range of
a
a
a
such that
∫
0
π
{
f
′
(
x
)
}
2
d
x
≥
f
(
π
2
)
.
\int_{0}^{\pi}\{f'(x)\}^{2}\ dx \geq f\left(\frac{\pi}{2}\right).
∫
0
π
{
f
′
(
x
)
}
2
d
x
≥
f
(
2
π
)
.
186
1
Hide problems
Today's calculation of Integral 186
For
a
>
0
,
a>0,
a
>
0
,
find
lim
a
→
∞
a
−
(
3
2
+
n
)
∫
0
a
x
n
1
+
x
d
x
(
n
=
1
,
2
,
⋯
)
.
\lim_{a\to\infty}a^{-\left(\frac{3}{2}+n\right) }\int_{0}^{a}x^{n}\sqrt{1+x}\ dx\ (n=1,\ 2,\ \cdots).
lim
a
→
∞
a
−
(
2
3
+
n
)
∫
0
a
x
n
1
+
x
d
x
(
n
=
1
,
2
,
⋯
)
.
185
1
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Today's calculation of Integral 185
Evaluate the following integrals. (1)
∫
0
π
4
d
x
1
+
sin
x
.
\int_{0}^{\frac{\pi}{4}}\frac{dx}{1+\sin x}.
∫
0
4
π
1
+
s
i
n
x
d
x
.
(2)
∫
4
3
2
d
x
x
2
x
−
1
.
\int_{\frac{4}{3}}^{2}\frac{dx}{x^{2}\sqrt{x-1}}.
∫
3
4
2
x
2
x
−
1
d
x
.
182
1
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Today's calculation of Integral 182
Find the area of the domain of the system of inequality
y
(
y
−
∣
x
2
−
5
∣
+
4
)
≤
0
,
y
+
x
2
−
2
x
−
3
≤
0.
y(y-|x^{2}-5|+4)\leq 0,\ \ y+x^{2}-2x-3\leq 0.
y
(
y
−
∣
x
2
−
5∣
+
4
)
≤
0
,
y
+
x
2
−
2
x
−
3
≤
0.
183
1
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Today's calculation of Integral 183
Let
n
≥
2
n\geq 2
n
≥
2
be integer. On a plane there are
n
+
2
n+2
n
+
2
points
O
,
P
0
,
P
1
,
⋯
P
n
O,\ P_{0},\ P_{1},\ \cdots P_{n}
O
,
P
0
,
P
1
,
⋯
P
n
which satisfy the following conditions as follows. [1]
∠
P
k
−
1
O
P
k
=
π
n
(
1
≤
k
≤
n
)
,
∠
O
P
k
−
1
P
k
=
∠
O
P
0
P
1
(
2
≤
k
≤
n
)
.
\angle{P_{k-1}OP_{k}}=\frac{\pi}{n}\ (1\leq k\leq n),\ \angle{OP_{k-1}P_{k}}=\angle{OP_{0}P_{1}}\ (2\leq k\leq n).
∠
P
k
−
1
O
P
k
=
n
π
(
1
≤
k
≤
n
)
,
∠
O
P
k
−
1
P
k
=
∠
O
P
0
P
1
(
2
≤
k
≤
n
)
.
[2]
O
P
0
‾
=
1
,
O
P
1
‾
=
1
+
1
n
.
\overline{OP_{0}}=1,\ \overline{OP_{1}}=1+\frac{1}{n}.
O
P
0
=
1
,
O
P
1
=
1
+
n
1
.
Find
lim
n
→
∞
∑
k
=
1
n
P
k
−
1
P
k
‾
.
\lim_{n\to\infty}\sum_{k=1}^{n}\overline{P_{k-1}P_{k}}.
lim
n
→
∞
∑
k
=
1
n
P
k
−
1
P
k
.
184
1
Hide problems
Today's calculation of Integral 184
(1) For real numbers
x
,
a
x,\ a
x
,
a
such that
0
<
x
<
a
,
0<x<a,
0
<
x
<
a
,
prove the following inequality.
2
x
a
<
∫
a
−
x
a
+
x
1
t
d
t
<
x
(
1
a
+
x
+
1
a
−
x
)
.
\frac{2x}{a}<\int_{a-x}^{a+x}\frac{1}{t}\ dt<x\left(\frac{1}{a+x}+\frac{1}{a-x}\right).
a
2
x
<
∫
a
−
x
a
+
x
t
1
d
t
<
x
(
a
+
x
1
+
a
−
x
1
)
.
(2) Use the result of
(
1
)
(1)
(
1
)
to prove that
0.68
<
ln
2
<
0.71.
0.68<\ln 2<0.71.
0.68
<
ln
2
<
0.71.
181
1
Hide problems
Today's calculation of Integral 181
For real number
a
,
a,
a
,
find the minimum value of
∫
0
π
2
∣
sin
2
x
1
+
sin
2
x
−
a
cos
x
∣
d
x
.
\int_{0}^{\frac{\pi}{2}}\left|\frac{\sin 2x}{1+\sin^{2}x}-a\cos x\right| dx.
∫
0
2
π
1
+
s
i
n
2
x
s
i
n
2
x
−
a
cos
x
d
x
.
180
1
Hide problems
Today's calculation of Integral 180
Let
a
n
a_{n}
a
n
be the area surrounded by the curves
y
=
e
−
x
y=e^{-x}
y
=
e
−
x
and the part of
y
=
e
−
x
∣
cos
x
∣
,
(
n
−
1
)
π
≤
x
≤
n
π
(
n
=
1
,
2
,
3
,
⋯
)
.
y=e^{-x}|\cos x|,\ (n-1)\pi \leq x\leq n\pi \ (n=1,\ 2,\ 3,\ \cdots).
y
=
e
−
x
∣
cos
x
∣
,
(
n
−
1
)
π
≤
x
≤
nπ
(
n
=
1
,
2
,
3
,
⋯
)
.
Evaluate
lim
n
→
∞
(
a
1
+
a
2
+
⋯
+
a
n
)
.
\lim_{n\to\infty}(a_{1}+a_{2}+\cdots+a_{n}).
lim
n
→
∞
(
a
1
+
a
2
+
⋯
+
a
n
)
.
179
1
Hide problems
Today's calculation of Integral 179
Evaluate the following integrals. (1) Meiji University
∫
1
e
e
(
log
x
)
2
x
d
x
.
\int_{\frac{1}{e}}^{e}\frac{(\log x)^{2}}{x}dx.
∫
e
1
e
x
(
l
o
g
x
)
2
d
x
.
(2) Tokyo University of Science
∫
0
1
7
x
3
+
23
x
2
+
21
x
+
15
(
x
2
+
1
)
(
x
+
1
)
2
d
x
.
\int_{0}^{1}\frac{7x^{3}+23x^{2}+21x+15}{(x^{2}+1)(x+1)^{2}}dx.
∫
0
1
(
x
2
+
1
)
(
x
+
1
)
2
7
x
3
+
23
x
2
+
21
x
+
15
d
x
.
178
1
Hide problems
Today's calculation of Integral 178
Let
f
(
x
)
f(x)
f
(
x
)
be a differentiable function such that
f
′
(
x
)
+
f
(
x
)
=
4
x
e
−
x
sin
2
x
,
f
(
0
)
=
0.
f'(x)+f(x)=4xe^{-x}\sin 2x,\ \ f(0)=0.
f
′
(
x
)
+
f
(
x
)
=
4
x
e
−
x
sin
2
x
,
f
(
0
)
=
0.
Find
lim
n
→
∞
∑
k
=
1
n
f
(
k
π
)
.
\lim_{n\to\infty}\sum_{k=1}^{n}f(k\pi).
lim
n
→
∞
∑
k
=
1
n
f
(
kπ
)
.
177
1
Hide problems
Today's calculation of Integral 177
On
x
y
xy
x
y
plane the parabola
K
:
y
=
1
d
x
2
(
d
:
p
o
s
i
t
i
v
e
c
o
n
s
t
a
n
t
n
u
m
b
e
r
)
K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)
K
:
y
=
d
1
x
2
(
d
:
p
os
i
t
i
v
e
co
n
s
t
an
t
n
u
mb
er
)
intersects with the line
y
=
x
y=x
y
=
x
at the point
P
P
P
that is different from the origin. Assumed that the circle
C
C
C
is touched to
K
K
K
at
P
P
P
and
y
y
y
axis at the point
Q
.
Q.
Q
.
Let
S
1
S_{1}
S
1
be the area of the region surrounded by the line passing through two points
P
,
Q
P,\ Q
P
,
Q
and
K
,
K,
K
,
or
S
2
S_{2}
S
2
be the area of the region surrounded by the line which is passing through
P
P
P
and parallel to
x
x
x
axis and
K
.
K.
K
.
Find the value of
S
1
S
2
.
\frac{S_{1}}{S_{2}}.
S
2
S
1
.
176
1
Hide problems
Today's calculation of Integral 176
Let
f
n
(
x
)
=
∑
k
=
1
n
sin
k
x
k
(
k
+
1
)
.
f_{n}(x)=\sum_{k=1}^{n}\frac{\sin kx}{\sqrt{k(k+1)}}.
f
n
(
x
)
=
∑
k
=
1
n
k
(
k
+
1
)
s
i
n
k
x
.
Find
lim
n
→
∞
∫
0
2
π
{
f
n
(
x
)
}
2
d
x
.
\lim_{n\to\infty}\int_{0}^{2\pi}\{f_{n}(x)\}^{2}dx.
lim
n
→
∞
∫
0
2
π
{
f
n
(
x
)
}
2
d
x
.
175
1
Hide problems
Today's calculation of Integral 175
Evaluate
∑
n
=
0
∞
1
(
2
n
+
1
)
2
2
n
+
1
.
\sum_{n=0}^{\infty}\frac{1}{(2n+1)2^{2n+1}}.
∑
n
=
0
∞
(
2
n
+
1
)
2
2
n
+
1
1
.
174
1
Hide problems
Today's calculation of Integral 174
Let
a
a
a
be a positive number. Assume that the parameterized curve
C
:
x
=
t
+
e
a
t
,
y
=
−
t
+
e
a
t
(
−
∞
<
t
<
∞
)
C: \ x=t+e^{at},\ y=-t+e^{at}\ (-\infty <t< \infty)
C
:
x
=
t
+
e
a
t
,
y
=
−
t
+
e
a
t
(
−
∞
<
t
<
∞
)
is touched to
x
x
x
axis. (1) Find the value of
a
.
a.
a
.
(2) Find the area of the part which is surrounded by two straight lines
y
=
0
,
y
=
x
y=0, y=x
y
=
0
,
y
=
x
and the curve
C
.
C.
C
.
173
1
Hide problems
Today's calculation of Integral 173
Find the function
f
(
x
)
f(x)
f
(
x
)
such that
f
(
x
)
=
cos
(
2
m
x
)
+
∫
0
π
f
(
t
)
∣
cos
t
∣
d
t
f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt
f
(
x
)
=
cos
(
2
m
x
)
+
∫
0
π
f
(
t
)
∣
cos
t
∣
d
t
for positive inetger
m
.
m.
m
.
172
1
Hide problems
Today's calculation of Integral 172
Evaluate
∫
−
1
0
1
+
x
1
−
x
d
x
.
\int_{-1}^{0}\sqrt{\frac{1+x}{1-x}}dx.
∫
−
1
0
1
−
x
1
+
x
d
x
.
171
1
Hide problems
Today's calculation of Integral 171
Evaluate
∫
0
1
x
2007
(
1
−
x
2
)
1003
d
x
.
\int_{0}^{1}x^{2007}(1-x^{2})^{1003}dx.
∫
0
1
x
2007
(
1
−
x
2
)
1003
d
x
.
170
1
Hide problems
Today's calculation of Integral 170
Let
a
,
b
a,\ b
a
,
b
be constant numbers such that
a
2
≥
b
.
a^{2}\geq b.
a
2
≥
b
.
Find the following definite 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
169
1
Hide problems
Today's calculation of Integral 169
(1) Let
f
(
x
)
f(x)
f
(
x
)
be the differentiable and increasing function such that
f
(
0
)
=
0.
f(0)=0.
f
(
0
)
=
0.
Prove that
∫
0
1
f
(
x
)
f
′
(
x
)
d
x
≥
1
2
(
∫
0
1
f
(
x
)
d
x
)
2
.
\int_{0}^{1}f(x)f'(x)dx\geq \frac{1}{2}\left(\int_{0}^{1}f(x)dx\right)^{2}.
∫
0
1
f
(
x
)
f
′
(
x
)
d
x
≥
2
1
(
∫
0
1
f
(
x
)
d
x
)
2
.
(2)
g
n
(
x
)
=
x
2
n
+
1
+
a
n
x
+
b
n
(
n
=
1
,
2
,
3
,
⋯
)
g_{n}(x)=x^{2n+1}+a_{n}x+b_{n}\ (n=1,\ 2,\ 3,\ \cdots)
g
n
(
x
)
=
x
2
n
+
1
+
a
n
x
+
b
n
(
n
=
1
,
2
,
3
,
⋯
)
satisfies
∫
−
1
1
(
p
x
+
q
)
g
n
(
x
)
d
x
=
0
\int_{-1}^{1}(px+q)g_{n}(x)dx=0
∫
−
1
1
(
p
x
+
q
)
g
n
(
x
)
d
x
=
0
for all linear equations
p
x
+
q
.
px+q.
p
x
+
q
.
Find
a
n
,
b
n
.
a_{n},\ b_{n}.
a
n
,
b
n
.
168
1
Hide problems
Today's calculation of Integral 168
Prove that
∑
n
=
1
∞
∫
1
n
+
1
1
n
∣
1
x
sin
π
x
∣
d
x
\sum_{n=1}^{\infty}\int_{\frac{1}{n+1}}^{\frac{1}{n}}{\left|\frac{1}{x}\sin \frac{\pi}{x}\right| dx}
∑
n
=
1
∞
∫
n
+
1
1
n
1
x
1
sin
x
π
d
x
diverge for
x
>
0.
x>0.
x
>
0.