Z 1 x dx p x3 1 2
[Pages:2]MIT Integration Bee Qualifying Exam Answers
1
log(2x) dx = log x + (log 2) log log x
x log x
2
1 dx = log 2
0 ex + 1
3
ee
log x
?
log (log x)
dx
=
1 (1
+
e2)
e
x
4
4
1
1+x
log
dx = log 4
0
1-x
5
1 dx = arctan (2x - 1)
x2 + (x - 1)2
6
x
xx
?
?
?
dx
=
x2/2
21 January 2020
7
sin4 x cos4 x(cos x + sin x)(cos x - sin x) dx = 1 sin5(x) cos5(x)
5
8 log(x2 + 1) dx = x log(x2 + 1) + 2 arctan(x) - 2x
9
2
cos2020(x) dx = 2-2019
2020
0
1010
10
2x + 1 dx = 1 ln (2x2 + 2x + 1)
2x2 + 2x + 1
2
11
1 arcsin(x)
1/ 2
x3
dx = 1/2
/2
12
sin(2x) cos(cos(x)) dx = 2(cos(1) + sin(1) - 1)
0
2
13
sin(sin(x) - x) dx = 0
0
14
1 +
x-1
2k0=108(k + 1)xk
2019 k=0
xk
dx
=
log(1
-
x2020)
15
2
1
dx = /4
0 tan 2020(x) + 1
16
x(1 - x)2020 dx = (1 - x)2022 - (1 - x)2021
2022
2021
17
sec4(x) tan(x)
1
dx = log
sec4(x) + 4
sec4(x) + 4
4
18 x2x(2 log(x) + 2) dx = x2x
1
19
1 - x2 dx = /4
0
20
x5e-x4
dx
=
/8
0
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