Theoretical Computer Science Cheat Sheet Page 6
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10Theoretical Computer Science Cheat Sheet
π
Calculus
Wallis’ identity:
Derivatives:
2 · 2 · 4 · 4 · 6 · 6 · · ·
π = 2 ·
d(cu)
du
d(u + v)
du
dv
d(uv)
dv
du
1 · 3 · 3 · 5 · 5 · 7 · · ·
1.
= c
,
2.
=
+
,
3.
= u
+ v
,
dx
dx
dx
dx
dx
dx
dx
dx
)
)
Brouncker’s continued fraction expansion:
du
dv
n
cu
v
u
d(u
)
du
d(u/v)
d(e
)
du
2
1
n 1
dx
dx
cu
4.
= nu
,
5.
=
,
6.
= ce
,
π
= 1 +
2
dx
dx
dx
v
dx
dx
4
2
3
2 +
5 2
2+
u
d(c
)
du
d(ln u)
1
du
7 2
2+
u
2+···
7.
= (ln c)c
,
8.
=
,
dx
dx
dx
u
dx
Gregrory’s series:
· · ·
d(sin u)
du
d(cos u)
du
π
1
1
1
1
= 1
+
+
9.
= cos u
,
10.
=
sin u
,
4
3
5
7
9
dx
dx
dx
dx
Newton’s series:
d(tan u)
du
d(cot u)
du
2
2
11.
= sec
u
,
12.
= csc
u
,
1 · 3
1
1
dx
dx
dx
dx
+ · · ·
π
=
+
+
2 · 3 · 2
2 · 4 · 5 · 2
6
3
5
2
d(sec u)
du
d(csc u)
du
13.
= tan u sec u
,
14.
=
cot u csc u
,
Sharp’s series:
dx
dx
dx
dx
(
)
d(arcsin u)
1
du
d(arccos u)
1
du
1
1
1
1
√
√
√
+ · · ·
π
15.
=
,
16.
=
,
=
1
+
· 3
· 5
· 7
dx
dx
dx
dx
6
1
2
3
2
2
1
u
1
u
3
3
3
3
d(arctan u)
1
du
d(arccot u)
1
du
Euler’s series:
17.
=
,
18.
=
,
2
2
dx
1 + u
dx
dx
1 + u
dx
2
+ · · ·
π
1
1
1
1
1
=
+
+
+
+
d(arcsec u)
1
du
d(arccsc u)
1
du
2
2
2
2
2
6
1
2
3
4
5
√
√
19.
=
,
20.
=
,
2
+ · · ·
π
1
1
1
1
1
dx
2
dx
dx
2
dx
u
1
u
u
1
u
=
+
+
+
+
2
2
2
2
2
8
1
3
5
7
9
d(sinh u)
du
d(cosh u)
du
2
· · ·
π
1
1
1
1
1
=
+
+
21.
= cosh u
,
22.
= sinh u
,
2
2
2
2
2
12
1
2
3
4
5
dx
dx
dx
dx
Partial Fractions
d(tanh u)
du
d(coth u)
du
2
2
23.
= sech
u
,
24.
=
csch
u
,
Let N (x) and D(x) be polynomial func-
dx
dx
dx
dx
tions of x.
We can break down
d(sech u)
du
d(csch u)
du
25.
=
sech u tanh u
,
26.
=
csch u coth u
,
N (x)/D(x) using partial fraction expan-
dx
dx
dx
dx
sion. First, if the degree of N is greater
d(arcsinh u)
1
du
d(arccosh u)
1
du
than or equal to the degree of D, divide
√
√
27.
=
,
28.
=
,
dx
dx
dx
dx
2
2
1 + u
u
1
N by D, obtaining
′
N (x)
N
(x)
d(arctanh u)
1
du
d(arccoth u)
1
du
= Q(x) +
,
29.
=
,
30.
=
,
D(x)
D(x)
2
2
dx
1
u
dx
dx
u
1
dx
′
where the degree of N
is less than that of
d(arcsech u)
1
du
d(arccsch u)
1
du
√
√
31.
=
,
32.
=
.
D. Second, factor D(x). Use the follow-
|u|
dx
2
dx
dx
2
dx
u
1
u
1 + u
ing rules: For a non-repeated factor:
Integrals:
′
N (x)
A
N
(x)
∫
∫
∫
∫
∫
=
+
,
(x
a)D(x)
x
a
D(x)
1.
cu dx = c
u dx,
2.
(u + v) dx =
u dx +
v dx,
[
]
∫
∫
∫
where
1
1
N (x)
n ̸ =
n
n+1
x
x
3.
x
dx =
x
,
1,
4.
dx = ln x,
5.
e
dx = e
,
A =
.
n + 1
x
D(x)
x=a
∫
∫
∫
dx
dv
du
For a repeated factor:
6.
= arctan x,
7.
u
dx = uv
v
dx,
∑
2
m 1
1 + x
dx
dx
′
N (x)
A
N
(x)
∫
∫
k
=
+
,
m
m k
(x
a)
D(x)
(x
a)
D(x)
8.
sin x dx =
cos x,
9.
cos x dx = sin x,
k=0
∫
∫
where
[
(
)]
k
1
d
N (x)
ln | cos x|,
cot x dx = ln | cos x|,
10.
tan x dx =
11.
A
=
.
k
k
k!
dx
D(x)
∫
∫
x=a
sec x dx = ln | sec x + tan x|,
csc x dx = ln | csc x + cot x|,
12.
13.
The reasonable man adapts himself to the
∫
√
world; the unreasonable persists in trying
x
x
2
2
to adapt the world to himself. Therefore
14.
arcsin
dx = arcsin
+
a
x
,
a > 0,
a
a
all progress depends on the unreasonable.
– George Bernard Shaw
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