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Formula sheet for Prelim 3 in Math 106 on Tuesday April 24, 2007
Antiderivatives
(inde±nite integrals)
±
x
p
dx
=
x
p
+1
p
+1
for
p
±
=

1
±
x

1
dx
=ln

x

±
e
rx
dx
=
e
r
Substitution.
Suppose
G
±
=
g
. Letting
u
=
f
(
x
),
du
=
f
±
(
x
)
dx
±
g
(
f
(
x
))
f
±
(
x
)
dx
=
±
g
(
u
)
du
=
G
(
u
)=
G
(
f
(
x
))
Integration by parts
±
f
(
x
)
g
±
(
x
)
dx
=
f
(
x
)
g
(
x
)

±
f
±
(
x
)
g
(
x
)
dx
Defnite integrals.
The area under the curve
f
between
a
and
b
is given by
±
b
a
f
(
x
)
dx
=
F
(
b
)

F
(
a
F
(
x
)

b
a
where
F
is ANY antiderivative.
If we rotate the part of the curve
f
over the interval [
a, b
] around the
x
axis, the result is a
solid oF revolution
with volume
V
=
²
b
a
πf
(
x
)
2
dx
.
The improper integral
±
∞
0
f
(
x
)
dx
= lim
b
→∞
±
f
(
x
)
dx
To ±nd the limit it is useful to know that if
p, q, r >
0 then as
x
→∞
e

,x

q
q
e

,
(ln
x
)

p
,
(ln
x
)
p
x

q
→
0
A probability density has
f
(
x
)
≥
0 and
²
f
(
x
)
dx
= 1. The mean
μ
=
EX
=
²
xf
(
x
)
dx
.
The second moment
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