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Today's Calculation Of Integral
2007 Today's Calculation Of Integral
169
Today's calculation of Integral 169
Today's calculation of Integral 169
Source: Osaka women university entrance exam 1985
January 15, 2007
calculus
integration
function
calculus computations
Problem Statement
(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
.
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