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CMIMC Integration Bee
2023 CMIMC Integration Bee
2023 CMIMC Integration Bee
Part of
CMIMC Integration Bee
Subcontests
(15)
15
1
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2023 Integration Bee Problem 15
∫
0
∞
(
1
−
e
−
π
/
x
2
)
2
d
x
\int_0^\infty \left(1-e^{-\pi/x^2}\right)^2\,\mathrm dx
∫
0
∞
(
1
−
e
−
π
/
x
2
)
2
d
x
Proposed by Vlad Oleksenko
14
1
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2023 Integration Bee Problem 14
∫
0
∞
e
−
⌊
x
⌋
(
1
+
{
x
}
)
d
x
\int_0^\infty e^{-\lfloor x \rfloor(1+\{x\})}\,\mathrm dx
∫
0
∞
e
−
⌊
x
⌋
(
1
+
{
x
})
d
x
Proposed by Vlad Oleksenko
13
1
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2023 Integration Bee Problem 13
∫
0
1
2
x
log
2
(
2
)
+
log
2
(
1
+
x
)
d
x
\int_0^1 2^{\sqrt x}\log^2(2)+\log^2(1+x)\,\mathrm dx
∫
0
1
2
x
lo
g
2
(
2
)
+
lo
g
2
(
1
+
x
)
d
x
Proposed by Thomas Lam
12
1
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2023 Integration Bee Problem 12
lim
n
→
∞
n
2
∫
0
1
x
n
e
−
x
log
(
x
)
d
x
\lim_{n\to\infty} n^2 \int_0^1 x^n e^{-x}\log(x)\,\mathrm dx
n
→
∞
lim
n
2
∫
0
1
x
n
e
−
x
lo
g
(
x
)
d
x
Proposed by Connor Gordon and Vlad Oleksenko
11
1
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2023 Integration Bee Problem 11
∫
−
1
1
1
(
8
+
x
2
)
1
−
x
2
d
x
\int_{-1}^1 \frac{1}{(8 + x^2)\sqrt{1-x^2}} \,\mathrm dx
∫
−
1
1
(
8
+
x
2
)
1
−
x
2
1
d
x
Proposed by Vlad Oleksenko
10
1
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2023 Integration Bee Problem 10
∫
1
3
3
arctan
(
x
)
log
2
(
x
)
x
d
x
\int_{\frac 1{\sqrt 3}}^{\sqrt 3} \frac{\arctan(x)\log^2(x)}{x}\,\mathrm dx
∫
3
1
3
x
arctan
(
x
)
lo
g
2
(
x
)
d
x
Proposed by Connor Gordon
9
1
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2023 Integration Bee Problem 9
∫
−
1
1
x
2022
cos
(
π
12
−
x
)
sin
(
π
12
+
x
)
d
x
\int_{-1}^1 x^{2022}\cos\left(\tfrac \pi {12}-x\right)\sin\left(\tfrac \pi{12}+x\right)\,\mathrm dx
∫
−
1
1
x
2022
cos
(
12
π
−
x
)
sin
(
12
π
+
x
)
d
x
Proposed by Michael Duncan, Connor Gordon, and Vlad Oleksenko
8
1
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2023 Integration Bee Problem 8
∫
−
10
10
∣
4
−
∣
3
−
∣
2
−
∣
1
−
∣
x
∣
∣
∣
∣
∣
d
x
\int_{-10}^{10}|4-|3-|2-|1-|x|||||\,\mathrm dx
∫
−
10
10
∣4
−
∣3
−
∣2
−
∣1
−
∣
x
∣∣∣∣∣
d
x
Proposed by Connor Gordon
7
1
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2023 Integration Bee Problem 7
∫
0
π
2
(
1
1
−
cos
(
x
)
−
2
x
2
)
d
x
\int_0^{\frac \pi 2} \left(\frac{1}{1-\cos(x)}-\frac{2}{x^2}\right)\,\mathrm dx
∫
0
2
π
(
1
−
cos
(
x
)
1
−
x
2
2
)
d
x
Proposed by Vlad Oleksenko
6
1
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2023 Integration Bee Problem 6
∫
0
2
e
x
(
x
4
+
8
x
3
+
18
x
2
+
16
x
+
5
)
d
x
\int_0^2 e^x(x^4+8x^3+18x^2+16x+5)\,\mathrm dx
∫
0
2
e
x
(
x
4
+
8
x
3
+
18
x
2
+
16
x
+
5
)
d
x
Proposed by Connor Gordon
5
1
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2023 Integration Bee Problem 5
∫
1
∞
1
x
x
2023
−
1
d
x
\int_1^\infty \frac{1}{x\sqrt{x^{2023}-1}}\,\mathrm dx
∫
1
∞
x
x
2023
−
1
1
d
x
Proposed by Connor Gordon
4
1
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2023 Integration Bee Problem 4
∫
0
∞
x
e
−
x
3
d
x
\int_0^\infty x e^{-\sqrt[3]{x}}\,\mathrm dx
∫
0
∞
x
e
−
3
x
d
x
Proposed by Connor Gordon
3
1
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2023 Integration Bee Problem 3
∫
0
π
4
cot
(
x
)
sin
(
x
)
d
x
\int_0^{\frac \pi 4} \cot(x)\sqrt{\sin(x)}\,\mathrm dx
∫
0
4
π
cot
(
x
)
sin
(
x
)
d
x
Proposed by Connor Gordon
2
1
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2023 Integration Bee Problem 2
∫
0
1
1
x
+
x
d
x
\int_0^1 \frac{1}{x+\sqrt x}\,\mathrm dx
∫
0
1
x
+
x
1
d
x
Proposed by Connor Gordon
1
1
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2023 Integration Bee Problem 1
∫
2
0
x
2
+
3
d
x
\int_2^0 x^2+3\,\mathrm dx
∫
2
0
x
2
+
3
d
x
Proposed by Connor Gordon