HIGH SCHOOL ROUND ONE
⇒
You will have two minutes to evaluate each of the fifteen definite
integrals that will be displayed one at a time on this screen. All
answers must be simplified. At the end of the two minutes, all hands
must go up and judges will grade your answers immediately. For each
correct answer, you will receive one raffle ticket to be entered for prizes
that will be drawn after dinner.
At most five participants will move to the finals – to be determined by
the total number of correct answers and tiebreaking criteria if
necessary. Everyone moving to the Finals will receive $25.
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INTEGRAL #1
READY,
GET SET,…
2:00
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B E E
INTEGRAL #1
∫ π/6
(
)
sin(2x) + cos(6x) dx
0
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INTEGRAL #1
∫ π/6
(
)
sin(2x) + cos(6x) dx
0
[
]π/6
cos(2x) sin(6x)
= −
+
2
6
0
=
1
4
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B E E
INTEGRAL #2
READY,
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2:00
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B E E
INTEGRAL #2
∫1
(x − 2)(x − 1)(x + 1)(x + 2) dx
0
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INTEGRAL #2
∫1
(x − 2)(x − 1)(x + 1)(x + 2) dx
0
∫1
(x2 − 4)(x2 − 1) dx
=
0
∫1
(x4 − 5x2 + 4) dx
=
0
]1
5x
38
x
−
+ 4x =
=
5
3
15
0
[
2 0 1 5
3
5
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B E E
INTEGRAL #3
READY,
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2:00
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B E E
INTEGRAL #3
∫1
xe2x dx
0
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INTEGRAL #3
∫1
xe2x dx
0
[
integrate by parts:
[
=
]
2x 1
xe
2
[
−
0
1
2
e
xe
=
−
2
4
2 0 1 5
U
of
du = dx
∫1
,
dv = e dx
]
v = 21 e2x
e2x dx
0
]
2x 1
2x
u=x
2x
0
S
e2 + 1
=
4
I NT E G RAT I ON
B E E
INTEGRAL #4
READY,
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2:00
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B E E
INTEGRAL #4
∫ 1 (√
)
√
4
5
5
x + x4 dx
0
2:00
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B E E
INTEGRAL #4
∫ 1 (√
)
√
4
5
5
x + x4 dx
0
∫1 (
=
5/4
x
4/5
+x
)
dx
0
[
=
]
9/5 1
9/4
4x
9
+
5x
9
0
= 1
2 0 1 5
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B E E
INTEGRAL #5
READY,
GET SET,…
2:00
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B E E
INTEGRAL #5
∫ π/4
0
sin x
dx
(1 + sin x)(1 − sin x)
2:00
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B E E
INTEGRAL #5
∫ π/4
0
sin x
dx
(1 + sin x)(1 − sin x)
∫ π/4
=
0
sin x
dx =
2
1 − sin x
∫ √2/2
=−
1
∫ π/4
0
sin x
dx
2
cos x
[
1
du u = cos x,
u2
]
du = − sin x dx
[ ]√2/2
√
1
=
=
2−1
u 1
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B E E
INTEGRAL #6
READY,
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2:00
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B E E
INTEGRAL #6
∫2 √
3
−3
x+3
dx
5
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B E E
INTEGRAL #6
∫2 √
3
−3
x+3
dx
5
∫1
=5
0
[
[
x+3
1/3
,
u dx u =
5
]
4/3 1
15u
=
4
=
]
1
du = dx
5
0
15
4
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B E E
INTEGRAL #7
READY,
GET SET,…
2:00
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B E E
INTEGRAL #7
∫ 4/π
3/π
1
1
1
sec tan dx
2
x
x
x
2:00
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B E E
INTEGRAL #7
∫ 4/π
3/π
1
1
1
sec tan dx
2
x
x
x
∫ π/4
=−
π/3
[
1
sec u tan u du u = ,
x
]
1
du = − 2 dx
x
[
]π/4
= − sec u
π/3
= 2−
2 0 1 5
√
2
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B E E
INTEGRAL #8
READY,
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2:00
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B E E
INTEGRAL #8
∫ ln 3
0
ex
dx
x
3
(1 + e )
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B E E
INTEGRAL #8
∫ ln 3
0
ex
dx
x
3
(1 + e )
∫4
=
[
u−3 du u = 1 + ex,
du = ex dx
]
2
[
]4
1
= − 2
2u 2
3
=
32
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B E E
INTEGRAL #9
READY,
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2:00
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B E E
INTEGRAL #9
∫ π/4
(tan3 x + 1)(tan2 x + 1) dx
0
2:00
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B E E
INTEGRAL #9
∫ π/4
(tan3 x + 1)(tan2 x + 1) dx
0
∫ π/4
(tan3 x + 1) sec2 x dx
=
0
∫1
=
[
(u3 + 1) du u = tan x,
]
du = sec2 x dx
0
[
4
u
=
+u
4
2 0 1 5
U
]1
0
of
5
=
4
S
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B E E
INTEGRAL #10
READY,
GET SET,…
2:00
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B E E
INTEGRAL #10
∫1
√
2/2
(
)2
1
1
1 + 2 dx
3
x
x
2:00
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B E E
INTEGRAL #10
∫1
√
2/2
(
)2
1
1
1 + 2 dx
3
x
x
[
1
1
=−
u2 dx u = 1 + 2 ,
2 3
x
[ 3 ]2
u
= −
6 3
∫2
]
2
du = − 3 dx
x
19
=
6
2 0 1 5
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B E E
INTEGRAL #11
READY,
GET SET,…
2:00
2 0 1 5
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B E E
INTEGRAL #11
∫ π2
√
sin x
√
dx
π2/4
x
2:00
2 0 1 5
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B E E
INTEGRAL #11
∫ π2
√
sin x
√
dx
π2/4
x
[
∫π
sin u du
=2
π/2
[
= − 2 cos u
u=
√
x,
]
1
du = √ dx
2 x
]π
π/2
= 2
2 0 1 5
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B E E
INTEGRAL #12
READY,
GET SET,…
2:00
2 0 1 5
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B E E
INTEGRAL #12
∫ √2 √
x4 − x2 dx
1
2:00
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B E E
INTEGRAL #12
∫ √2 √
x4 − x2 dx
1
∫ √2 √ √
∫ √2 √
=
x2 x2 − 1 dx =
x x2 − 1 dx
1
1
=
2
[
=
1
∫1
√
[
u du
2
u = x − 1,
]
du = 2x dx
0
]
3/2 1
u
3
2 0 1 5
=
0
U
of
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3
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B E E
INTEGRAL #13
READY,
GET SET,…
2:00
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B E E
INTEGRAL #13
∫ 13
6
1
dx
5/3
(x − 5)
2:00
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B E E
INTEGRAL #13
∫ 13
6
1
dx
5/3
(x − 5)
∫8
u−5/3
=
[
u = x − 5,
]
du = dx
1
]8
[
3
= − 2/3
2u
1
9
=
8
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B E E
INTEGRAL #14
READY,
GET SET,…
2:00
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B E E
INTEGRAL #14
∫3
0
x
√
dx
x+1
2:00
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B E E
INTEGRAL #14
∫3
0
x
√
dx
x+1
[
u=
∫2
=
1
√
x + 1,
]
2
(u2 − 1) · 2u
du
u
[
∫2
u3
2
−u
= 2 (u − 1) du = 2
3
1
2 0 1 5
2u du = dx
u = x + 1,
U
of
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]2
1
8
=
3
I NT E G RAT I ON
B E E
INTEGRAL #15
READY,
GET SET,…
2:00
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B E E
INTEGRAL #15
∫3
1
x2016 − x2014
dx
2015
2014
x
+x
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B E E
INTEGRAL #15
∫3
1
x2016 − x2014
dx
2015
2014
x
+x
∫3
=
1
x2014(x2 − 1)
dx
x2014(x + 1)
∫3
x2014(x + 1)(x − 1)
=
dx
2014
x (x + 1)
1
[ 2
]3
∫3
x
= (x − 1) dx =
−x = 2
2
1
1
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THANKS FOR PLAYING
LET’S EAT!
(YOU HAVE TWO MINUTES TO FINISH YOUR FOOD)
THE FINAL ROUND BEGINS AFTER DINNER
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