微積分定理,微積分公式

1. 常用數學與微積分公式定理 richwang ( 1 / 7 )
常用數學公式
常 用 數 學 公 式
a x
dt
t
x
special cases
b xy x y x r x
x
y
x y
x
r
) ln ( )
ln( ) , ln( ) , ln( )
) ln( ) ln ln ln( ) ln
ln( ) ln ln
= >
= = −∞ ∞ = +∞
= + = ⋅
= −
z1
0
1 0 0:
a e e e j
b e x
c e y y y
e y y y
j
x
y
y
) . , , cos sin
) exp( )
) exp(ln )
ln ln(exp( ))
ln
1 0
2 718281828 1≈ = = +
=
= ⇔ =
= ⇔ =
θ
θ θ
sin( ) sin cos cos sin
sin( ) sin cos cos sin
cos( ) cos cos sin sin
cos( ) cos cos sin sin
sin cos sin( ) sin( )
cos sin sin( ) sin( )
cos cos cos( ) cos( )
sin sin cos( ) cos( )
( ) /
(
A B A B A B
A B A B A B
A B A B A B
A B A B A B
A B A B A B
A B A B A B
A B A B A B
A B A B A B
assume
x A B
y A B
then
A x y
B x
+ = ⋅ + ⋅
− = ⋅ − ⋅
+ = ⋅ − ⋅
− = ⋅ + ⋅
⋅ = + + −
⋅ = + − −
⋅ = + + −
⋅ = − + − −
= +
= −
RST
= +
=
1
2
1
2
1
2
1
2
2
−
RST
+ = ⋅
+
⋅
−
− = ⋅
+
⋅
−
+ = ⋅
+
⋅
−
− = − ⋅
+
⋅
−
y
x y
x y x y
x y
x y x y
x y
x y x y
x y
x y x y
) /
sin sin sin cos
sin sin cos sin
cos cos cos cos
cos cos sin sin
2
2
2 2
2
2 2
2
2 2
2
2 2
sin sin cos
cos cos sin cos sin
cos
cos
, sin
cos
sin cos
sin tan sec , cos cot csc
2 2
2 2 1 1 2
1 2
2
1 2
2
1
1 1
2 2 2 2
2 2 2 2
2 2 2 2 2 2
θ θ θ
θ θ θ θ θ
θ
θ
θ
θ
θ θ
θ θ θ θ θ θ
=
= − = − = −
=
+
=
−
⇔ + =
÷ ⇒ + = ÷ ⇒ + =
a
a a
b g
b gc h b gc h

2. 常用數學與微積分公式定理 richwang ( 2 / 7 )
常用微分公式
常 用 微 分 公 式
d f g g df f dg
du du d u u
dC
dx
dC C constant
d x C d x d x d x C
( )
( )
( : )
( ) ( )
= +
+ = +
= ⇔ =
+ = ⇔ = +
1 2 1 2
0 0
de
dx
e de e dx
d x
dx x u
du d u
x
x x x
= ⇔ =
= ⇔ =
ln
ln
1 1
d x
dx
n x d x n x dx
d x
dx
x d x xdx
d
dx x x
d
x x
dx
n
n n n
= ⇔ =
= ⇔ =
F
HG I
KJ =
−
⇔
F
HG I
KJ =
−
− −1 1
2
2
2 2
2 2
1 1 1 1
d xy ydx xdy
d x y m x y dx n y x dy
m ydx n xdy
d x y
x y
d
y
x
xdy ydx
x
d
x
y
ydx xdy
y
m n m n n m
m n
m n
( )
( )
= +
= ⋅ + ⋅
∴ ⋅ + ⋅ =
F
HG I
KJ =
−
F
HG I
KJ =
−
− −
− −
c h 1 1
1 1
2
2
d x
dx
x d x x dx
d x
dx
x d x x dx
d x
dx
x d x x dx
d x
dx
x d x x dx
d x
dx
x x d x x x dx
d x
dx
x x d x x x dx
sin
cos sin cos
cos
sin cos sin
tan
sec tan sec
cot
csc cot csc
sec
sec tan sec sec tan
csc
csc cot csc csc cot
= ⇔ =
= − ⇔ = −
= ⇔ =
= − ⇔ = −
= ⇔ =
= − ⇔ = −
2 2
2 2
d x
dx x
dx
x
d x
d x
dx x
dx
x
d x
d x
dx x x
dx
x x
d x
d x
dx x
dx
x
d x
d x
dx x
dx
x
d x
d x
dx x x
dx
x x
d x
sin
sin
tan
tan
sec
sec
cos
cos
cot
cot
csc
csc
−
−
−
−
−
−
−
−
−
−
−
−
=
−
⇔
−
=
=
+
⇔
+
=
=
−
⇔
−
=
=
−
−
⇔
−
−
=
=
−
+
⇔
−
+
=
=
−
−
⇔
−
−
=
1
2 2
1
1
2 2
1
1
2 2
1
1
2 2
1
1
2 2
1
1
2 2
1
1
1 1
1
1 1
1
1 1
1
1 1
1
1 1
1
1 1

3. 常用數學與微積分公式定理 ( 3 / 7 )
微積分定理與公式
微 積 分 定 理 與 公 式
d
dx
f x g x
d
dx
f x
d
dx
g x
d
dx
f g g
d
dx
f f
d
dx
g f g f g f g
f
g
f g f g
g
g x
y y u u u x
dy
dx
dy
du
du
dx
( ) ( ) ( ) ( )
( )
( ) ( )
± = ±
⋅ = ⋅ + ⋅ ⇔ ⋅ ′= ′⋅ + ⋅ ′
L
NM O
QP
′
=
′⋅ − ⋅ ′
⇒ ≠
= = = ⋅
2
0
chain rule : if and then
d F x
dx
f x f x dx F x C
d
dx
f t dt f x a
d
dx
f x dx f x d f x dx f x dx
f x dx d f x dx
d f x
dx
dx f x C d f x f x C C
y y x dy y dx
u u x y du
u
x
dx
u
y
dy
u dv u v v du
a
x
( )
( ) ( ) ( )
( ) ( ) , :
( ) ( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( ) ( : )
( )
( , )
= ⇒ = +
=
= => =
=
= + => = +
= = ′⋅
= =
∂
∂
⋅ +
∂
∂
⋅
R
S|
T|
⋅ = ⋅ − ⋅
z
z
z z
z
zz
zz
constant.
integral constant
( integral by parts )
( ) ( )
d
dx
f x t dt f x b x
db
dx
f x p x
d p
dx x
f x t dt
p x
b x
p x
b x
( , ) , ( ) , ( ) ( , )
( ) ( )
z z= − +
∂
∂
b g b g
dC
dx
d x
dx
n x
n
n
=
= −
0
1
de
dx
e
de
dx
a e
da
dx
a a
x
x chain rule
a x
a x
x
x
= ⎯ →⎯⎯⎯ =
= ⋅ln

4. 常用數學與微積分公式定理 ( 4 / 7 )
微積分定理與公式
微 積 分 定 理 與 公 式
d x
dx
x
d x
dx
x
d x
dx
x
d x
dx
x
d x
dx
x
d x
dx
x
d x
dx
x
d x
dx
x
d x
dx
x x
d x
dx
x x
d x
dx
x x
d x
dx
x
chain rule
chain rule
chain rule
chain rule
chain rule
chain rule
sin
cos
sin
cos
cos
sin
cos
sin
tan
sec
tan
sec
cot
csc
cot
csc
sec
sec tan
sec
sec tan
csc
csc cot
csc
csc
= ⎯ →⎯⎯⎯ = ⋅
= − ⎯ →⎯⎯⎯ = − ⋅
= ⎯ →⎯⎯⎯ = ⋅
= − ⎯ →⎯⎯⎯ = − ⋅
= ⋅ ⎯ →⎯⎯⎯ = ⋅ ⋅
= − ⋅ ⎯ →⎯⎯⎯ = − ⋅ ⋅
ω
ω ω
ω
ω ω
ω
ω ω
ω
ω ω
ω
ω ω ω
ω
ω ω
2 2
2 2
cotω x
d x
dx x
d x
dx x
d x
dx x
d x
dx x
d x
dx x
d x
dx x
d x
dx x
d x
dx x
d x
dx x x
d x
dx x
chain rule
chain rule
chain rule
chain rule
chain rule
sin sin
( )
cos cos
( )
tan tan
( )
cot cot
( )
sec sec
(
− −
− −
− −
− −
− −
=
−
⎯ →⎯⎯⎯ =
−
=
−
−
⎯ →⎯⎯⎯ =
−
−
=
+
⎯ →⎯⎯⎯ =
+
=
−
+
⎯ →⎯⎯⎯ =
−
+
=
−
⎯ →⎯⎯⎯ =
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
1
1 1
1
1 1
1
1 1
1
1 1
1
1
1
ω ω
ω
ω ω
ω
ω ω
ω
ω ω
ω
ω
ω
ω
ω
x
d x
dx x x
d x
dx x x
chain rule
)
csc csc
( )
2
1
2
1
2
1
1
1
1
1
−
=
−
−
⎯ →⎯⎯⎯ =
−
−
− −
Binomial formula
x y C x y C x y
Leibniz s formula
d
dx
f g f g C f g
where C
n
k
n
k n k
n
k
n
k
n
k n k
k
n
k
n
n k k
n
n
n
k
n
k
n
k n k
k
n
:
' :
!
! !
( ) ( ) ( )
+ = =
⋅ = ⋅ =
=
F
HG I
KJ =
−
=
−
=
−
=
−
∑ ∑
∑
b g
b g b g
b g
0 0
0

5. 常用數學與微積分公式定理 ( 5 / 7 )
微積分定理與公式
* Taylor’s series expansion :
f x
f a
n
x a
f a
f a
x a
f a
x a
f a
x a
where n n n n
f(x)is an infinitely differentiable function
e
x
n
x x x
x
x
n
x
x
n
n
n
x
n
n
n n
n
( )
( )
!
( )
( )
( )
!
( )
( )
!
( )
( )
!
( )
! .
.
! ! ! !
sin
( )
( )!
( )
( )
= ⋅ −
= +
′
− +
′′
− + − +
= − −
= = + + + +⋅⋅⋅⋅⋅
=
−
+
= −
=
∞
=
∞
+
=
∞
∑
∑
∑
0
2
3
3
0
2 3
2 1
0
1 2 3
1 2 2 1
1
1 2 3
1
2 1
b gb g b gb g
some important expansions:
3 5 7
2
0
2 4 6
2 3
3 5 7
1
2
1
2 4 6
1 1
1
2
1 2
3
! ! !
cos
( )
( )! ! ! !
! !
+ − +⋅⋅⋅⋅⋅
=
−
= − + − +⋅⋅⋅⋅⋅
+ = + +
−
+
− −
+
=
∞
∑
x x
x
x
n
x x x
x px
p p
x
p p p
x
n n
n
p
b g b g b gb g
If then
d
dx
F x f x f x dx F x C
d
dx
Cb g b g b g b g= = + ⇒ =
L
NM O
QPz, 0
x dx
n
x C n
dx
x
x C
du
u
u C
n n
=
+
+ ≠ −
= + ⇒ = +
+
z
z z
1
1
11
b g
ln ln
e dx
a
e C
a dx
a
a C a e e
a x a x
x x x a x a
x
= ⋅ +
= ⋅ + ⇒ = =
z
z ⋅
1
1
ln
ln lne j
1
1
1
1
1
1
1
1
1
1
1
1
2
1
2
1
2
1
2
1
2
1
2
1
−
= +
−
−
= +
+
= +
−
+
= +
−
= +
−
−
= +
z z
z z
z z
− −
− −
− −
x
dx x C
x
dx x C
x
dx x C
x
dx x C
x x
dx x C
x x
dx x C
sin cos
tan cot
sec csc

6. 常用數學與微積分公式定理 ( 6 / 7 )
微積分定理與公式
e bx dx
e
a b
a bx b bx C
e bx dx
e
a b
a bx b bx C
a x
a x
a x
a x
cos cos sin
sin sin cos
=
+
⋅ + ⋅ +
=
+
⋅ − ⋅ +
z
z
2 2
2 2
sin cos
cos sin
tan ln sec sec tan
cot ln sin csc cot
sec ln sec tan sec tan
csc ln csc cot csc cot
sin cos
cos sin
tan ln sec
cot ln sin
x dx x C
x dx x C
x dx x C x dx x C
xdx x C xdx x C
x dx x x C x dx x C
x dx x x C x dx x C
x dx x C
x dx x C
xdx x C
x dx
z
z
z z
z z
z z
z z
z
z
z
z
= − +
= +
= + ⇒ = +
= + ⇒ = − +
= + + ⇒ = +
= − + + ⇒ = − +
= − ⋅ +
= ⋅ +
= +
=
2
2
2
2
1
1
1
1
1
1
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω
ω x C
xdx x x C
xdx x x C
+
= + +
= − + +
z
z
sec ln sec tan
csc ln csc cot
ω
ω
ω ω
ω
ω
ω ω
1
1