Practice Test For Extraordinary People And a Few Species of
Swallow And Dolphin; Also “Calculus And You“: A Self-Help
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 warm-up:
(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 x-axis.
(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 1-6, 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
This
preview
has intentionally blurred sections.
Sign up to view the full version.
(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 Half-way 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?

This is the end of the preview.
Sign up
to
access the rest of the document.
- Spring '09
- Calculus, Trigonometry, dx, Trig Subst
-
Click to edit the document details