Integrals
Derivatives
• Volume: Suppose A(x) is the cross-sectional area
of the solid S perpendicular to the x-axis, then
the volume of S is given by
•
d
(sinh x) = cosh x
dx
d
(cosh x) = sinh x
dx
• Inverse Trigonometric Functions:
b
Z
V =
A(x) dx
a
• Work: Suppose f (x) is a force function. The
work in moving an object from a to b is given by:
Z
b
d
1
(sin−1 x) = √
dx
1 − x2
d
−1
(csc−1 x) = √
dx
x x2 − 1
d
−1
(cos−1 x) = √
dx
1 − x2
d
1
(sec−1 x) = √
dx
x x2 − 1
d
1
(tan−1 x) =
dx
1 + x2
d
−1
(cot−1 x) =
dx
1 + x2
f (x) dx
W =
a
Z
•
• If f is a one-to-one differentiable function with
0
inverse function f −1 and f (f −1 (a)) 6= 0, then
the inverse function is differentiable at a and
1
dx = ln |x| + C
x
Z
•
tan x dx = ln | sec x| + C
1
0
(f −1 ) (a) =
f
0
(f −1 (a))
Z
•
sec x dx = ln | sec x + tan x| + C
Hyperbolic and Trig Identities
Z
•
ax
a dx =
+C
ln a
x
for a 6= 1
• Hyperbolic Functions
• Integration by Parts:
Z
Z
u dv = uv − v du
sinh(x) =
ex − e−x
2
csch(x) =
1
sinh x
• Arc Length Formula:
cosh(x) =
ex + e−x
2
sech(x) =
1
cosh x
tanh(x) =
sinh x
cosh x
coth(x) =
cosh x
sinh x
Z bp
1 + [f 0 (x)]2 dx
L=
a
• cosh2 x − sinh2 x = 1
• sin2 x = 12 (1 − cos 2x)
• cos2 x = 12 (1 + cos 2x)
• sin(2x) = 2 sin x cos x
• sin A cos B = 12 [sin(A − B) + sin(A + B)]
• sin A sin B = 12 [cos(A − B) − cos(A + B)]
• cos A cos B = 12 [cos(A − B) + cos(A + B)]
• Alternating Series Test: If the alternating
∞
X
series
(−1)n−1 bn satisfies
Series
• nth term test for divergence: If lim an does
n→∞
not exist or if lim an 6= 0, then the series
n→∞
∞
X
an
n=1
n=1
(i) 0 < bn+1 ≤ bn
(ii) lim bn = 0
n→∞
is divergent.
∞
X
1
is convergent if p > 1
• The p-series:
np
n=1
and divergent if p ≤ 1.
• Geometric: If |r| < 1 then
∞
X
arn =
n=0
a
1−r
• The Integral Test: Suppose f is a continuous,
positive, decreasing function on [1, ∞) and let
an = f (n). Then
Z ∞
(i) If
f (x) dx is convergent,
1
then
Z
(ii) If
∞
X
an is convergent.
n=1
∞
f (x) dx is divergent,
1
then
∞
X
for all n
then the series is convergent.
• The Ratio Test
an+1 = L < 1, then the series
(i) If lim n→∞
an ∞
X
an is absolutely convergent.
n=1
an+1 an+1 = ∞,
(ii) If lim = L > 1 or lim n→∞
n→∞
an an ∞
X
an is divergent.
then the series
n=1
an+1 = 1, the Ratio Test is
(iii) If lim n→∞
an inconclusive.
• Maclaurin Series: f (x) =
∞
X
f (n) (0)
n=0
an is divergent.
n=1
P
• TheP
Comparison Test: Suppose that
an
and
bn are series with positive terms.
P
(i) If
bP
n is convergent and an ≤ bn for all n,
then
an is also convergent.
P
(ii) If
bP
n is divergent and an ≥ bn for all n,
then
an is also divergent.
• The
P Limit
PComparison Test: Suppose that
an and
bn are series with positive terms. If
lim
n→∞
an
=c
bn
n!
xn
• Taylor’s Inequality If |f (n+1) (x)| ≤ M for
|x − a| ≤ d, then the remainder Rn (x) of the
Taylor series satisfies the inequality
|Rn (x)| ≤
M
|x − a|n+1
(n + 1)!
for |x − a| ≤ d
• Some Power Series
◦ ex =
∞
X
xn
n=0
∞
X
◦ sin x =
◦ cos x =
where c is a finite number and c > 0, then either
both series converge or both diverge.
R=∞
n!
(−1)n
n=0
∞
X
(−1)n
n=0
◦ ln(1 + x) =
∞
X
x2n+1
(2n + 1)!
R=∞
x2n
(2n)!
R=∞
(−1)n−1
n=1
xn
n
R=1
∞
◦
X
1
=
xn
1−x
n=0
R=1