常用微积分公式

sin( A − B) = sin A⋅ cos B − cos A⋅ sin B
cos( A + B) = cos A⋅ cos B − sin A⋅ sin B
cos( A − B) = cos A⋅ cos B + sin A⋅ sin B
sin A ⋅cos B = 1 sin( A + B) + sin( A − B) 2
n=0 n!
1! 2! 3!
∑ sin x = ∞ (−1)n x2n+1 = x − x3 + x5 − x7 + ⋅⋅⋅⋅⋅
n=0 (2n + 1)!
3! 5! 7!
b g b g b gb g ∑ cos x = ∞ (−1)n x2n = 1− x2 + x4 − x6 + ⋅⋅⋅⋅⋅
n=0 (2n)!
d sinx = cosx ⇔ d sinx= cosx dx dx
d cosx = − sinx ⇔ d cosx= − sinx dx dx
d tanx = sec2 x ⇔ d tanx= sec2 x dx dx
d cot x = − csc2 x ⇔ d cot x= − csc2 x dx dx
b g ∑ ∑ n
n
x + y n = Ckn x k yn−k = Ckn xn−k y k
k =0
k=0
Leibniz' s formula :
b g b g ∑ dn
dxn
f ⋅g =
n
f ⋅ g (n) = Ckn f g (k ) (n−k )
k =0
FHG IKJ b g where Ckn =
=
1 1+ x2
⇔
dx 1+ x2
=
d
tan−1x
d sec−1x = 1
⇔
dx = d sec−1x
dx
x x2 −1
x x2 −1
d cos−1x = −1
⇔
dx
1− x2
−dx = d cos−1x 1− x2
d cot −1x = −1 dx 1+ x2
⇔
−dx 1+ x2
= d cot−1x
dx
dx
d csc x = − csc x ⋅ cot x ⎯c⎯hain⎯r⎯ule→ d cscω x = −ω ⋅ cscω x ⋅ cot ω x
dx
dx
d sin−1 x = 1
dx
1− x2
⎯c⎯hain⎯r⎯ule→ d sin−1 ω x =
ω
dx
1− (ω x)2
d cos−1 x = −1
1!
2!
3!
b gb g b gb g where n! = n n −1 n − 2 " 2 1 .
f(x) is an infinitely differentiable function.
some important expansions :
∑ ex = ∞ xn = 1+ x + x2 + x3 + ⋅⋅⋅⋅⋅
d xn = n xn−1 ⇔ d xn = n xn−1dx dx
d x2 = 2x ⇔ d x2 = 2x dx
dx
F I F I d HG KJ HG KJ dx
1 x
= −1 x2
⇔
d1 x
=
−1 x2 dx
d ex = ex ⇔ d ex = exdx dx
d ln x = 1 ⇔ 1 du = d ln u
dx
dx
d tan x = sec2 x ⎯c⎯hain⎯r⎯ule→ d tanω x = ω ⋅ sec2ω x
dx
dx
d cot x = − csc2 x ⎯c⎯hain⎯r⎯ule→ d cot ω x = −ω ⋅ csc2ω x
dx
dx
d sec x = sec x ⋅ tan x ⎯c⎯hain⎯r⎯ule→ d secω x = ω ⋅ secω x ⋅ tanω x
dx
1+ (ω x)2
d sec−1 x = 1
⎯c⎯hain⎯r⎯ule→ d sec−1 ω x =
1
dx
x x2 −1
dx
x (ω x)2 − 1
d csc−1 x = −1
⎯c⎯hain⎯r⎯ule→ d csc−1 ω x =
−1
dx
x x2 −1
dx
x (ω x)2 − 1
Binomial formula :
ln( x ) = ln x − ln y y
a) e1 ≈ 2.718281828 , e0 = 1 , e jθ = cosθ + j sinθ b) ex = exp ( x) c) eln y = y ⇔ exp (ln y) = y
ln ey = y ⇔ ln (exp( y)) = y
sin( A + B) = sin A ⋅cos B + cos A⋅ sin B
cos A⋅ sin B = 1 sin( A + B) − sin( A − B) 2
cos A ⋅cos B = 1 cos( A + B) + cos( A − B) 2
sin A⋅ sin B = − 1 cos( A + B) − cos( A − B)
RST RST assume
2 x = A+B y = A − B then
z z cot x dx = ln sin x + C ⇒ csc2x dx = − cot x + C
z z sec x dx = ln sec x + tan x + C ⇒ sec2ω x dx = 1 tanω x + C
常用數學與微積分公式定理 ( 1 / 7 )
z a) ln x = x d t 1t
常用數學公式 ( x > 0)
special cases : ln(1) = 0 , ln(0) = −∞, ln(∞) = + ∞
b) ln(x y) = ln x + ln y
ln(xr ) = r ⋅ ln x
bn ≠ 1g
z z n +1 dx = ln x + C ⇒
du = ln u + C
x
u
z eaxdx = 1 ⋅ eax + C za
a x dx = 1 ⋅ a x + C ⇒ e j a x = eln ax = ex⋅lna
ln a
z 1 dx = sin−1x + C
1− x2
1
z 1+ x2
A = (x + y) / 2 B = (x + y) / 2
sin x + sin y = 2 ⋅ sin x + y ⋅ cos x − y
2
2
sin x − sin y = 2 ⋅cos x + y ⋅ sin x − y
2
2
cos x + cos y = 2 ⋅ cos x + y ⋅ cos x − y
( C : integral constant )
dx
Ry = y(x) S|T|u = u(x, y)
dy = y′⋅dx
du = ∂u ⋅dx + ∂u ⋅d y
∂x
∂y
z z u ⋅dv = u ⋅v − v ⋅du ( integral by parts )
z b g b g z d
b( x)
d csc−1x = −1
⇔
−dx = d csc−1x
dx
x x2 −1
x x2 −1
常用微分公式
常用數學與微積分公式定理 ( 3 / 7 )
微積分定理與公式
d f (x) ± g(x) = d f (x) ± d g(x)
dx
dx
dx
d f ⋅g = g ⋅ d f + f ⋅ d g ⇔ f ⋅g ′= f ′⋅ g + f ⋅ g′
f (x, t)dt = f
x, b(x)
db
−
f
x, p(x)
dp
+
∂ b( x) f (x, t)dt
dx p(x)
dx
dx
p( x) ∂x
Leabharlann Baidu
dC = 0 dx d xn = n xn−1 dx
d ex = ex ⎯c⎯hain⎯r⎯ule→ dx d a x = ln a ⋅ a x dx
d eax = a eax dx
d secx = sec x tan x ⇔ d secx = sec x tan x dx dx
d cscx = − csc x cot x ⇔ d cscx = − csc x cot x dx dx
d sin−1x = 1
⇔
dx
1− x2
dx = d sin−1x 1− x2
d
tan−1x dx
2
2
cos x − cos y = −2 ⋅ sin x + y ⋅ sin x − y
2
2
sin 2θ = 2 sinθ cosθ
cos2θ = cos2θ − sin2θ = 2 cos2θ − 1 = 1− 2 sin2θ
b g cos2 θ = 1+ cos2θ , sin2 θ = 1− cos2θ ⇔ sin2 θ + cos2 θ = 1 " a
n k
=
n! k! n−k !
微積分定理與公式
常用數學與微積分公式定理 ( 5 / 7 )
* Taylor’s series expansion :
∑∞
f (x) =
f (n) (a) ⋅ (x − a)n
n=0 n !
= f (a) + f ′(a) (x − a) + f ′′(a) (x − a)2 + f (3) (a) (x − a)3 + ""
d
x
f (t) dt = f (x)
, a : constant.
z z dx a
d f (x) dx = f (x) => d f (x) dx = f (x) dx
z dx
f (x) dx = d f (x) dx
z z d f (x) dx = f (x) + C => d f (x) = f (x) + C
dx
=
tan−1x
+
C
z 1 dx = sec−1x + C
x x2 −1
z − 1 dx = cos−1x + C
1− x2
−
z1+
1 x2
dx
=
cot −1x
+
C
z − 1 dx = csc−1x + C
x x2 −1
微積分定理與公式
常用數學與微積分公式定理 ( 6 / 7 )
z eax
cosbx dx
=
ea x a2 + b2
a ⋅ cosbx + b ⋅ sinbx + C
z eax
sin bx dx
=
ea x a2 + b2
a ⋅ sinbx − b ⋅ cosbx
+C
z sin x dx = − cos x + C
z cos x dx = sin x + C
z z tan x dx = ln sec x + C ⇒ sec2x dx = tan x + C
dx
1− x2
⎯c⎯hain⎯r⎯ule→ d cos−1 ω x =
−ω
dx
1− (ω x)2
d tan−1 x = 1
dx
1+ x2
d cot −1 x = −1 dx 1+ x2
⎯c⎯hain⎯r⎯ule→ ⎯c⎯hain⎯r⎯ule→
d tan−1 ω x = ω
dx
1+ (ω x)2
d cot−1 ω x = −ω
dx
dx
dx
LM OPf
′ =
f ′ ⋅ g − f ⋅g′
N Qg
g2
⇒ g(x) ≠ 0
chain rule : if y = y(u) and u = u(x) then dy = dy ⋅ du
dx du dx
z d F(x) = f (x) ⇒ f (x) dx = F(x) + C
zdx
微積分定理與公式
常用數學與微積分公式定理 ( 4 / 7 )
微積分定理與公式
d sin x = cos x ⎯c⎯hain⎯r⎯ule→ d sinω x = ω ⋅ cosω x
dx
dx
d cos x = − sin x ⎯c⎯hain⎯r⎯ule→ d cosω x = −ω ⋅ sinω x
2! 4! 6!
1+ x p = 1+ px + p p − 1 x2 + p p − 1 p − 2 x3 + ""
2!
3!
z b g b g b g b g LM OP If d F x = f x , then f x dx = F x + C
N Q dx
⇒ d C=0 dx
z xndx = 1 xn+1 + C
dx x
u
d (x y) = y dx + x dy
c h d xm yn = m⋅ xm−1yndx + n ⋅ yn−1xmdy
∴m ⋅
ydx
+n⋅
xdy
=
d(xmyn) xm−1 y n−1
FHG IKJ d
y x
=
x dy − y dx x2
FHG IKJ d
x y
=
y dx − x dy y2
cb g h cb g h 2
2
a ÷ sin2 θ ⇒ 1+ tan2 θ = sec2 θ ,
a ÷ cos2 θ ⇒ 1+ cot2 θ = csc2 θ
常用數學公式
常用數學與微積分公式定理 ( 2 / 7 )
常用微分公式
d ( f g) = g df + f dg
d u1 + d u2 = d (u1 + u2 ) d C = 0 ⇔ d C = 0 ( C : constant) dx d(x + C) = d x ⇔ d x = d(x + C)