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Jozsef Wildt International Math Competition
2009 Jozsef Wildt International Math Competition
W. 7
Prove this integration inequality
Prove this integration inequality
Source: 2009 Jozsef Wildt International Mathematical Competition
April 17, 2020
integration
logarithms
inequalities
calculus
Problem Statement
If
0
<
a
<
b
0<a<b
0
<
a
<
b
then
∫
a
b
(
x
2
−
(
a
+
b
2
)
2
)
ln
x
a
ln
x
b
(
x
2
+
a
2
)
(
x
2
+
b
2
)
d
x
>
0
\int \limits_a^b \frac{\left (x^2-\left (\frac{a+b}{2} \right )^2\right )\ln \frac{x}{a} \ln \frac{x}{b}}{(x^2+a^2)(x^2+b^2)} dx > 0
a
∫
b
(
x
2
+
a
2
)
(
x
2
+
b
2
)
(
x
2
−
(
2
a
+
b
)
2
)
ln
a
x
ln
b
x
d
x
>
0
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