高等数学一常用公式表
常用公式表(一)
1。乘法公式
22
(1)(a +b )=a 2+2ab +b 2 (2)(a -b )=a 2-2ab +b 2
(3)a 2-b 2=(a +b )(a -b )
(4)a 3+b 3=(a +b )(a 2-ab +b 2) (5)a 3-b 3=(a -b )(a 2+ab +b 2)
2、指数公式:
(1)a (4)a
=1(a ≠0) (2)a a =a
n
n
-p
=
1a
n p
m
(
3)a n
a a
m n
=
n
m n
m n m +n
(5)a
m
÷a =
n
n
=a
m -n
(6)(a m )
=a
(7)(ab )
(
10a ⎛a ⎫
=a b (8) ⎪=n
b ⎝b ⎭
n
(
9)2
=a
1
N
=a
(11)
b =log
1a
=a
-1
(
12b
=a 2
3、指数与对数关系:
(1)若a
b
=N
,则
b
(3)若e =N ,则b =ln
4、对数公式:
a
b
N
(2)若10
N
=N
,则b =lg
(1)
log
a
a =b
, ln e b
=b
(2)log a 1=0, ln 1=0
=ln N
(3)a (5)a
b
log aN
=N
ln a
b ln a M =e (6)ln MN =ln M +ln N (7)ln =ln M -ln N
N
n 1
(8)ln M =n ln M (
9)ln =ln M
n
,e ln N
=N
(4)log a N
(1)sin 2α
(4)
sin αcos α
=tan α
5、三角恒等式: 22
+cos α=1 (2)1+tan α(3)1+cot 2α=csc 2α (5)
=
1sin α
cos αsin α
=cot α
=sec α
2
1
(6)cot α
=
1cos α
=
tan α
(7)csc α
(8)sec α
6. 倍角公式:
(1)sin (3)cos
(1)sin
2
2α=2sin αcos α
2
2
(2)tan 2α
2
=
2tan α1-tan α
2
2
2α=cos α-sin α=2cos α-1=1-2sin α
=
sin α1+cos α
7. 半角公式(降幂公式):
α
2=
1-cos α
2
(2)cos 2
α
2
=
1+cos α
2
(3)tan
α
2
=
1-cos αsin α
(1)若x=siny,则y=arcsinx (2)若x=cosy
,则y=arccosx (3)若x=tany,则y=arctanx (4)若x=coty,则y=arccotx
10、函数定义域求法:
1
(1)分式中的分母不能为0,(a α≠0) (2)负数不能开偶次方, (
a
α≥0)
g o l a N (3)对数中的真数必须大于0,( N>0)
(4)反三角函数中arcsinx ,arccosx 的x 满足:(-1≤x ≤1) (5)上面数种情况同时在某函数出现时,此时应取其交集。
11、直线形式及直线位置关系:
(1) 直线形式:点斜式:y -
y 0=k (x -x 0)
=x -x 1
斜截式:y=kx+b
y -y 1
两点式:y 2-y 1x 2-x 1
(2)直线关系:l 1:y =k 1x +b 1 l 2:y =k 2x +b 2 平行:若l 1//l 2,则k 1
=k 2 垂直:若l 1⊥l 2,则k 1⋅k 2=-1
(1)sin (α±β)=sin αcos β±cos αsin β(2)cos (α±β)=cos αcos β
(3)tan (α
±β)=
tan α±tan β1 tan α⋅tan β
sin αsin β
14、奇偶性及反函数:
(1)奇函数:f (-x )=-f (x ) (图象关于原点对称)
(2)偶函数:f (-x )=f (x ) (图象关于y 轴对称) (3)性质:奇±奇=奇; 奇±偶=非奇非偶; 偶±偶=偶
奇⨯÷奇=偶; 奇⨯÷偶=奇; 偶⨯÷偶=偶
(4)y =f (x )与y =
f
-1
(x )关于直线y
=x
对称
常用公式表(二)
1、求导法则:
(1)(u
'(cu ) (3)
'
+v )=u '+v '
(2)(u
'
-v )=u '-v '
=c u '
'(uv ) (4)
=u 'v +u v '
'u 'v -uv '
⎛u ⎫
(5) ⎪= 2
v ⎝v ⎭
2、基本求导公式:
'
(C ) (1)
=0
x
''
()()x =ax a =a (2) (3)
a
a -1
x
x
x
ln a
'
()e =e 5log x '= (4)()()
a
1x ln a
(6)(ln x )'=
x
1
'
(sin x )(7)
=cos x
2
'
(cos x ) (8)
=-sin x 1sin x
2
2
(9)(tan x )'
=
1cos x
2
=sec x (10)(cot x )'=-
'(csc x ) (12)
=-csc x
'
(sec x )(11)
=sec x ⋅tan x =-csc x ⋅cot x
(13)(
arcsin x )'
==
11+x
2
(14)(
arccos x )'=- (16)(arc cot x )'=-
11+x
(15)(arctan x )'
2
3、微分
(1)函数的微分:dy =y 'dx =f '(x )dx
(2)近似计算:|Δx|很小时,f (x 0+∆x )≈f (x 0)+
4、基本积分公式
kdx
(1)⎰
=kx +C
f '(x 0)∆x
1x
(2)
+C
⎰
x
x dx =
a
1a +1
x
x
a +1
+C
(3)
⎰
=ln x +C
(4)
⎰a
x
dx =
a
x
ln a
e
(5)⎰
dx =e +C
(6)⎰
2
sin xdx =-cos x +C
cos
(7)⎰
xdx =sin x +C
(8)
⎰sec
xdx =
⎰cos
1-x
1
2
x
=tan x +C
(9)
⎰csc
2
xdx =
⎰sin
1
2
x
=-cot x +C
(10)
⎰
2
=arcsin x +C
x (11)
5、定积分公式:
⎰1+
1
2
dx =arctan x +C
(1)⎰a
b
b
f (x )dx =
a
⎰
b a
f (t )dt (2)⎰f
a
a
(x )dx
c
=0
b
(3)⎰a f (x )dx =-⎰b f (x )dx (4)⎰a f (x )dx =⎰a f (x )dx +⎰c f (x )dx
(5)若f (x )是[-a,a]的连续奇函数,则⎰-a
(6)若f (x )是[-a,a]的连续偶函数,则⎰-a
6、积分定理:
(1)⎡
⎢⎰a
⎣
x
b
a
f (x )dx =0
a
a
f (x )dx =2⎰f (x )dx
'b (x )
⎡⎤(2)⎢⎰f (t )dt ⎥=f [b (x )]b '(x )-f [a (x )]a '(x )'
⎤⎣a (x )⎦f (t )dt =f (x ) ⎥⎦
b
(3)若F (x )是f (x )的一个原函数,则⎰a
7. 积分表
(1)⎰sec
xdx =ln sec x +tan x +C
f (x )dx =F (x )
b a
=F (b )-F (a )
(2)⎰csc
xdx =ln csc x -cot x +C
(3)⎰
1a
2
+x
2
=
1a
arctan
x a
+C
=2
12a
ln
(4)⎰
1a
2
-x
2
=arcsin
x a
+C
(5)⎰
1x -a
2
x -a x +a
+C
8.积分方法
(一)凑微分:(y 'dx =dy )
(1)e x dx =de x ; (2)cos xdx =d sin x ; (3)sin xdx =-d cos x ; (4)dx =(5)x
dx =
a
1a
d (ax +b )1
2
1a +1
(9(1)f (x )=
cos x sin x
1
=d arcsin x ; (10)dx =d arctan x 21+x dx
a +1
; (6)
1x
dx =d ln x ; (7)
1
2
dx =d tan x ; (8)
dx =-d cot x
(二)换元法:
ax +b
2
;设:
ax +b =t
则:dx =x '(t )dt
=a cos t ; dx
=a cos tdt
(2)f (x )=
f (x )=f (x )=
a
a
2
-x
2
2
;设:x =a sin t x -a
2
;设:x =a sec t udv =uv -
=a tan t , dx =a
sec t tan tdt =a sec t , dx =a sec tdt
2
+x
2
;设:x =a tan t
(三)分部积分法:⎰
9.全微分及隐函数
(1)全微分:dz =z 'x dx +z 'y dy
⎰vdu
(2)隐函数:若F (x , y )=0, 则:y '=
∂z
dy dx
=-
F x 'F y '
F y 'F x '∂z
(3)隐函数:若F (x , y , z )=0, 则:=-; =-
∂x F z '∂y F z '