Theoretical Computer Science Cheat Sheet
Calculus Cont. Finite Calculus
62.
dx
x
√
x
2
− a
2
=
1
a
arccos
a
|x|
,a>0, 63.
dx
x
2
√
x
2
± a
2
= ∓
√
x
2
± a
2
a
2
x
,
64.
xdx
√
x
2
± a
2
=
x
2
± a
2
, 65.
√
x
2
± a
2
x
4
dx = ∓
(x
2
+ a
2
)
3/2
3a
2
x
3
,
66.
dx
ax
2
+ bx + c
=
1
√
b
2
− 4ac
ln
2ax + b −
√
b
2
− 4ac
2ax + b +
√
b
2
− 4ac
, if b
2
> 4ac,
2
√
4ac − b
2
arctan
2ax + b
√
4ac − b
2
, if b
2
< 4ac,
67.
dx
√
ax
2
+ bx + c
=
1
√
a
ln
2ax + b +2
√
a
ax
2
+ bx + c
, if a>0,
1
√
−a
arcsin
−2ax − b
√
b
2
− 4ac
, if a<0,
68.
ax
2
+ bx + cdx=
2ax + b
4a
ax
2
+ bx + c +
4ax − b
2
8a
dx
√
ax
2
+ bx + c
,
69.
xdx
√
ax
2
+ bx + c
=
√
ax
2
+ bx + c
a
−
b
2a
dx
√
ax
2
+ bx + c
,
70.
dx
x
√
ax
2
+ bx + c
=
−1
√
c
ln
2
√
c
√
ax
2
+ bx + c + bx +2c
x
, if c>0,
1
√
−c
arcsin
bx +2c
|x|
√
b
2
− 4ac
, if c<0,
71.
x
3
x
2
+ a
2
dx =(
1
3
x
2
−
2
15
a
2
)(x
2
+ a
2
)
3/2
,
72.
x
n
sin(ax) dx = −
1
a
x
n
cos(ax)+
n
a
x
n−1
cos(ax) dx,
73.
x
n
cos(ax) dx =
1
a
x
n
sin(ax) −
n
a
x
n−1
sin(ax) dx,
74.
x
n
e
ax
dx =
x
n
e
ax
a
−
n
a
x
n−1
e
ax
dx,
75.
x
n
ln(ax) dx = x
n+1
ln(ax)
n +1
−
1
(n +1)
2
,
76.
x
n
(ln ax)
m
dx =
x
n+1
n +1
(ln ax)
m
−
m
n +1
x
n
(ln ax)
m−1
dx.
Difference, shift operators:
∆f(x)=f(x +1)− f(x),
E
f(x)=f(x +1).
Fundamental Theorem:
f(x)=∆F (x) ⇔
f(x)δx = F (x)+C.
b
a
f(x)δx =
b−1
i=a
f(i).
Differences:
∆(cu)=c∆u, ∆(u + v)=∆u +∆v,
∆(uv)=u∆v +
E
v∆u,
∆(x
n
)=nx
n−1
,
∆(H
x
)=x
−1
, ∆(2
x
)=2
x
,
∆(c
x
)=(c − 1)c
x
, ∆
x
m
=
x
m−1
.
Sums:
cu δx = c
uδx,
(u + v) δx =
uδx+
vδx,
u∆vδx= uv −
E
v∆uδx,
x
n
δx =
x
n+1
m+1
,
x
−1
δx = H
x
,
c
x
δx =
c
x
c−1
,
x
m
δx =
x
m+1
.
Falling Factorial Powers:
x
n
= x(x − 1) ···(x − n +1),n>0,
x
0
=1,
x
n
=
1
(x +1)···(x + |n|)
,n<0,
x
n+m
= x
m
(x − m)
n
.
Rising Factorial Powers:
x
n
= x(x +1)···(x + n −1),n>0,
x
0
=1,
x
n
=
1
(x − 1) ···(x −|n|)
,n<0,
x
n+m
= x
m
(x + m)
n
.
Conversion:
x
n
=(−1)
n
(−x)
n
=(x − n +1)
n
=1/(x +1)
−n
,
x
n
=(−1)
n
(−x)
n
=(x + n − 1)
n
=1/(x − 1)
−n
,
x
n
=
n
k=1
n
k
x
k
=
n
k=1
n
k
(−1)
n−k
x
k
,
x
n
=
n
k=1
n
k
(−1)
n−k
x
k
,
x
n
=
n
k=1
n
k
x
k
.
x
1
= x
1
= x
1
x
2
= x
2
+ x
1
= x
2
− x
1
x
3
= x
3
+3x
2
+ x
1
= x
3
− 3x
2
+ x
1
x
4
= x
4
+6x
3
+7x
2
+ x
1
= x
4
− 6x
3
+7x
2
− x
1
x
5
= x
5
+15x
4
+25x
3
+10x
2
+ x
1
= x
5
− 15x
4
+25x
3
− 10x
2
+ x
1
x
1
= x
1
x
1
= x
1
x
2
= x
2
+ x
1
x
2
= x
2
− x
1
x
3
= x
3
+3x
2
+2x
1
x
3
= x
3
− 3x
2
+2x
1
x
4
= x
4
+6x
3
+11x
2
+6x
1
x
4
= x
4
− 6x
3
+11x
2
− 6x
1
x
5
= x
5
+10x
4
+35x
3
+50x
2
+24x
1
x
5
= x
5
− 10x
4
+35x
3
− 50x
2
+24x
1