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Jozsef Wildt International Math Competition
2019 Jozsef Wildt International Math Competition
W. 66
Prove this inequality as the bound of the integration
Prove this inequality as the bound of the integration
Source: 2019 Jozsef Wildt International Math Competition
May 20, 2020
integration
inequalities
trigonometry
inverse trigonometric function
calculus
Problem Statement
If
0
<
a
≤
b
0 < a \leq b
0
<
a
≤
b
then
2
3
tan
−
1
(
2
(
b
2
−
a
2
)
(
a
2
+
2
)
(
b
2
+
2
)
)
≤
∫
a
b
(
x
2
+
1
)
(
x
2
+
x
+
1
)
(
x
3
+
x
2
+
1
)
(
x
3
+
x
+
1
)
d
x
≤
4
3
tan
−
1
(
(
b
−
a
)
3
a
+
b
+
2
(
1
+
a
b
)
)
\frac{2}{\sqrt{3}}\tan^{-1}\left(\frac{2(b^2 - a^2)}{(a^2+2)(b^2+2)}\right)\leq \int \limits_a^b \frac{(x^2+1)(x^2+x+1)}{(x^3 + x^2 + 1) (x^3 + x + 1)}dx\leq \frac{4}{\sqrt{3}}\tan^{-1}\left(\frac{(b-a)\sqrt{3}}{a+b+2(1+ab)}\right)
3
2
tan
−
1
(
(
a
2
+
2
)
(
b
2
+
2
)
2
(
b
2
−
a
2
)
)
≤
a
∫
b
(
x
3
+
x
2
+
1
)
(
x
3
+
x
+
1
)
(
x
2
+
1
)
(
x
2
+
x
+
1
)
d
x
≤
3
4
tan
−
1
(
a
+
b
+
2
(
1
+
ab
)
(
b
−
a
)
3
)
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