Practice Test For Extraordinary People And a Few Species of
Swallow And Dolphin; Also “Calculus And You“: A SelfHelp
Guide
David Imberti
April 6, 2010
1
A Small Practice Test
The following have their answers worked out below (so, spoiler warning!), consider these as a warmup:
(1)
x
x
2
+1
dx
(2)
cos
(
θ
)
cos
(
sin
(
θ
))
d
θ
(3)
sec
4
(
θ
)
(4)
sin
4
(
θ
)
(5)
1
√
x
2
+2
x
dx
(6)
x
3
x
2

1
dx
Compute the following integrals (take your time, relax) (I have the solutions on my person, ask me if
you’re unsure):
(1)
ln
(
x
)
dx
(2)
arcsin
(
x
)
dx
(3)
x
4
+1
x
3
+
x
2

x

1
dx
(4)
(
y
2

y
)

1
dy
(5)
θ
tan
2
(
θ
)
d
θ
(6)
1
√
x
+1+
√
x
dx
(7)
n
= 3, approximate
f
(
x
) =
1
x
between 1 and 2 with the Trapezoidal rule.
(8) Connect the following dots: (

2
,
0)
,
(

2
,
2)
,
(

1
,
2)
,
(

1
,
1)
,
(1
,
1)
,
(1
,
3)
,
(3
,
3)
,
(3
,
0)
,
(

2
,
0). Find the
coordinates of the centroid of the block region inside these points.
(9) Find the surface area by rotating
x
=
1
3
(
y
2
+ 2)
3
2
,
1
≤
y
≤
2 about the xaxis.
(10) Is
∞
1
ln

x
+ 1


ln

x

dx
convergent or divergent?
You might also want to take a look at:
p. 448 25,73 p. 518 21,23,33,18,56,57,2,30,41,46,47 p. 549 34 p. 561 3ab, 12 p. 694 16, 19 p. 759 16, 8,
27
Or old exams.
2
Calculus and You
Q: OMIGOSH, THERE’S THIS INTEGRAL AND I DON’T KNOW HOW TO
. . .
WHAT’S THIS
. . .
I
DON’T EVEN
A: O.k., don’t panic, grab your towel.
Remember, you’ve got only a handful of methods to integrate with that you need to remember:
(0) It’s an integral you already know.
1
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(1) Direct Substitution.
(2) Trigonometric identities.
(3) Trigonometic substitution.
(4) Partial Fractions.
(5) Integration by Parts.
If you want to go about this methodically, I would go down the list in the order I’ve listed.
Q: COULD YOU EXPLAIN EVERYTHING IN THAT LIST?
A: Sure, what specifically?
2.1
0. Integrals We Know: You’re Halfway There!
Q: WHAT INTEGRALS SHOULD I JUST MEMORIZE?
A: O.k., I’m going to talk a little bit about the more complicated integrals we’ve already done. (you
should know things like
cos
(
θ
)
d
θ
=
sin
(
θ
)
,
e
x
dx
=
e
x
,
x
a
dx
=
1
a
+1
x
a
+1
(for
a
=

1, else
x

1
dx
=

x

2
))
Here are the common integrals that have funky forms that you might forget/should memorize:
(Common 1):
1
√
1

x
2
dx
=
arcsin
(
x
) +
C
(Common 2):
1
1+
x
2
dx
=
arctan
(
x
) +
C
(Common 3):
sec
(
θ
)
d
θ
=
ln

sec
(
θ
) +
tan
(
θ
)

+
C
(Common 4):
tan
(
θ
)
d
θ
=

ln

cos
(
θ
)

+
C
Also, these are common trigonometric integrals, but they’re easy to forget when you’re chugging down in
the mood of trigonometric substitution:
(Common 5):
sec
2
(
θ
)
d
θ
=
tan
(
θ
) +
C
(Common 6):
sec
(
θ
)
tan
(
θ
)
d
θ
=
sec
(
θ
) +
C
2.2
1. Substituting Out For a Better You
Q: HOW DO I KNOW WHAT TO SUBSTITUTE FOR?
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 Spring '09
 Calculus, Trigonometry, dx, Trig Subst

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