hmmt february 2020 integration bee finalshmmt february 2020 integration bee finals. solution 22...
TRANSCRIPT
![Page 1: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/1.jpg)
HMMT February 2020 Integration Bee Finals
Edward Jin
February 15, 2020Sponsored by Five Rings Capital
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 2: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/2.jpg)
Problem 0
Evaluate the following Integral:∫ 1
01 dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 3: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/3.jpg)
Solution 0
Answer:1
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 4: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/4.jpg)
Problem 1
Evaluate the following Integral:∫ π/2
0cos(x) sin−1(cos(x)) dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 5: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/5.jpg)
Solution 1
Answer:1
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 6: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/6.jpg)
Problem 2
Evaluate the following Integral:∫ π/4
0
tan(x) sec2(x) dx√2− tan2 x
dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 7: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/7.jpg)
Solution 2
Answer: √2− 1
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 8: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/8.jpg)
Problem 3
Evaluate the following Integral:∫tan−1(x)
x2dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 9: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/9.jpg)
Solution 3
Answer:
−1
2log(x2 + 1) + log x − tan−1(x)
x
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 10: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/10.jpg)
Problem 4
Evaluate the following Integral:∫ 1
0sin−1(
√x) dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 11: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/11.jpg)
Solution 4
Answer:π
4
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 12: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/12.jpg)
Problem 5
Evaluate the following Integral:
∫ 1
0
3
√x
3
√x
3
√x 3√· · · dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 13: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/13.jpg)
Solution 5
Answer:2
3
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 14: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/14.jpg)
Problem 6
Evaluate the following Integral:∫ 2π
0cos(x) cos(2x) cos(3x) cos(4x) cos(5x) cos(6x) dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 15: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/15.jpg)
Solution 6
Answer:0
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 16: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/16.jpg)
Problem 7
Evaluate the following Integral:∫sinx(x) (log sin x + x cot x) dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 17: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/17.jpg)
Solution 7
Answer:sinx(x) + C
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 18: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/18.jpg)
Problem 8
Evaluate the following Integral:∫sin(x)esec(x)
cos2(x)dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 19: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/19.jpg)
Solution 8
Answer:esec x + C
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 20: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/20.jpg)
Problem 9
Evaluate the following Integral:∫ 2π
0
(sin(3x)
sin(x)
)3
dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 21: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/21.jpg)
Solution 9
Answer:14π
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 22: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/22.jpg)
Problem 10
Evaluate the following Integral:∫sinh2 x dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 23: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/23.jpg)
Solution 10
Answer:1
4sinh(2x)− x
2+ C
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 24: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/24.jpg)
Problem 11
Evaluate the following Limit:
limN→∞
∫ π/2
0
sin(Nx)
sin xdx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 25: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/25.jpg)
Solution 11
Answer:π
2
Edward Jin
HMMT February 2020 Integration Bee Finals
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Problem 12
Evaluate the following Integral:∫log(1 + x2) dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 27: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/27.jpg)
Solution 12
Answer:x log(1 + x2)− 2x + 2 tan−1(x) + C
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 28: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/28.jpg)
Problem 13
Evaluate the following Integral:∫sec(x) cosh(x)(cosh(x) tan(x) + 2 sinh(x)) dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 29: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/29.jpg)
Solution 13
Answer:sec x cosh2 x + C
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 30: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/30.jpg)
Problem 14
Evaluate the following Integral:∫ e
0W (x) dx
where W (x) is the Lambert-W function, defined as the inverse off (x) = xex (i.e. W (x)eW (x) = x).
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 31: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/31.jpg)
Solution 14
Answer:e − 1
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 32: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/32.jpg)
Problem 15
Evaluate the following Integral:∫exxe
x
(log x +
1
x
)dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 33: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/33.jpg)
Solution 15
Answer:xe
x+ C
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 34: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/34.jpg)
Problem 16
Evaluate the following Integral:∫sin(1/x)
x3dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 35: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/35.jpg)
Solution 16
Answer:cos(1/x)
x− sin(1/x) + C
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 36: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/36.jpg)
Problem 17
Evaluate the following Integral:∫sin4 x + cos4 x dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 37: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/37.jpg)
Solution 17
Answer:3
4x +
1
16sin(4x) + C
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 38: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/38.jpg)
Problem 18
Evaluate the following Integral:∫ 2π
0cos10 x dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 39: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/39.jpg)
Solution 18
Answer:63
128π
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 40: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/40.jpg)
Problem 19
Evaluate the following Integral:∫ π/2
0
dx
sin x + cos x
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 41: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/41.jpg)
Solution 19
Answer:√
2 tanh−11√2
=
√2
2log
(1 +√
2
1−√
2
)
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 42: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/42.jpg)
Problem 20
Evaluate the following Integral:∫ √1 + x2 dx
Edward Jin
HMMT February 2020 Integration Bee Finals
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Solution 20
Answer:1
2sinh−1(x) +
1
2x√
1 + x2 + C
or1
2log(x +
√1 + x2) +
1
2x√
1 + x2 + C
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 44: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/44.jpg)
Problem 21
Evaluate the following Integral:∫ 1
0
(1
2+
x
3+
x2
8+
x3
40+ · · ·+ xn
n!(n + 2)+ · · ·
)dx
where the sum is infinite.
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 45: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/45.jpg)
Solution 21
Answer:e − 2
Edward Jin
HMMT February 2020 Integration Bee Finals
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Problem 22
Define the Gamma function as
Γ(x) =
∫ ∞0
e−ttx−1 dt.
The Gamma function additionally satisfies the property
Γ(x)Γ(1− x) = π csc(πz).
Given the above information, evaluate the following Integral:∫ 1
0log Γ(x)dx
Edward Jin
HMMT February 2020 Integration Bee Finals
![Page 47: HMMT February 2020 Integration Bee FinalsHMMT February 2020 Integration Bee Finals. Solution 22 Answer: 1 2 log(2ˇ) Edward Jin HMMT February 2020 Integration Bee Finals. Problem 23](https://reader034.vdocuments.us/reader034/viewer/2022051607/603a59a4704f6c3b614d0293/html5/thumbnails/47.jpg)
Solution 22
Answer:1
2log(2π)
Edward Jin
HMMT February 2020 Integration Bee Finals
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Problem 23
Evaluate the following Integral:∫dx
x2/3 + x4/3
Edward Jin
HMMT February 2020 Integration Bee Finals
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Solution 23
Answer:3 tan−1( 3
√x) + C
Edward Jin
HMMT February 2020 Integration Bee Finals
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Problem 24
Evaluate the following Integral:∫ ∞0
x
(x2 + 1)(a2x2 + 1)dx
for positive a.
Edward Jin
HMMT February 2020 Integration Bee Finals
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Solution 24
Answer:log a
a2 − 1
Edward Jin
HMMT February 2020 Integration Bee Finals