Math 19B
Midterm 2
Study Aid
November 16, 2010
PreChapter 7.1  Stu
ff
you Should Know!
•
Basic Antiderivatives.
The following is the minimum list of antiderivatives you should have either
memorized or written on your review card (
C
and
c
are constants):
c dx
=
cx
+
C
x
n
dx
=
x
n
+1
n
+ 1
+
C
(
n
=
−
1)
1
x
dx
= ln

x

+
C
e
x
dx
=
e
x
+
C
sin
x dx
=
−
cos
x
+
C
cos
x dx
= sin
x
+
C
sec
2
x dx
= tan
x
+
C
f
(
x
)
f
(
x
)
dx
= ln

f
(
x
)

+
C
1
x
2
+
a
2
dx
=
1
a
arctan
x
a
+
C
1
√
1
−
x
2
dx
= arcsin
x
+
C
•
Usubstitution
: For indefinite integrals the
u
subs formula may be expressed:
f
(
g
(
x
))
g
(
x
)
dx
=
f
(
u
)
du
where
u
=
g
(
x
). In the definite case the formula reads:
b
a
f
(
g
(
x
))
g
(
x
)
dx
=
g
(
b
)
g
(
a
)
f
(
u
)
du.
Chapter 7.1  Integration by parts.
Formula.
•
Integration by parts formula.
f
(
x
)
g
(
x
)
dx
=
f
(
x
)
g
(
x
)
−
g
(
x
)
f
(
x
)
dx
or, in the more catchy form:
u dv
=
uv
−
v du.
1
Remarks.
•
ILATE.
When considering the product of two functions in an integration by parts problem, we need to
choose one function to be
u
and one to be
dv
. For your ‘first attempt’ choose
u
by ‘order of appearance’
in the ‘ILATE’ sequence:
I
−
I
nverses (trig inverses such as arcsin
x
etc.)
L
−
L
ogarithms (usually ln
x
)
A
−
A
polynomial (either
x
,
x
2
or
x
3
.)
T
−
T
rig function (sin(
x
), cos(2
x
), etc.)
E
=
E
xponential (
e
x
,
e
3
x
, etc.).
•
Multiply by 1.
Remember you can write
f
(
x
) as 1
×
f
(
x
). This trick is commonly used in the ‘
I
’ or
‘
L
’ cases of ILATE; notably in calculating
ln
x dx
or
arcsin
x dx.
•
Multiple integration by parts.
If you choose
u
=
x
2
in the ‘
A
’ case of ILATE you may have to do
integration by parts (IBP) twice. Similarly, if you’re forced to choose
u
=
x
3
, you may have to do IBP
three times!
In the ‘
T
’ or ‘
E
’ cases of ILATE you’ll probably have do IBPs twice and use the trick that the integral
you get the second time around is exactly what you started with.
If you haven’t seen this type of
example or have no idea what I’m going on about (!) try:
e
4
x
sin(2
x
)
dx.
Chapter 7.2  Trigonometric integrals.
Formula.
•
Here are all the trig identities you should need:
sin
2
x
+ cos
2
x
= 1
sin
2
x
=
1
2
(1
−
cos(2
x
))
tan
2
x
+ 1 = sec
2
x
cos
2
x
=
1
2
(1 + cos(2
x
))
1 + cot
2
x
= csc
2
x
sin(2
x
) = 2 sin
x
cos
x.
Techniques.
•
sin
m
x
cos
n
x dx
.
1. If
m
is odd, factor out one sin
x
and use sin
2
+ cos
2
= 1 to express the rest of the integral in terms
of cos
x
. Then make the usubs,
u
= cos
x
.
2. If
n
is odd, factor out one cos
x
and use sin
2
+ cos
2
= 1 to express the rest of the integral in terms
of sin
x
. Then make the usubs,
u
= sin
x
.
3. If
m
and
n
are both even use the half angle (‘power reducing’) identities:
sin
2
x
=
1
2
(1
−
cos(2
x
))
and
cos
2
x
=
1
2
(1 + cos(2
x
))
.
•
tan
m
x
sec
n
x dx
.
Page 2
1. If
n
is even (think s
E
c =
E
ven), factor out one sec
2
x
and use tan
2
+1 = sec
2
x
to express the
rest of the integral in terms of tan
x
. Then make the usubs,
u
= tan
x
.