高数常用公式
平方立方:
(1)a -b =(a +b )(a -b ) (2)a +2ab +b =(a +b ) (3)a -2ab +b =(a -b )
3
3
2
2
2
2
2
2
2
2
2
2
(4)a +b =(a +b )(a -ab +b ) (5)a -b =(a -b )(a +ab +b ) (6)a +3a b +3ab +b =(a +b ) (7)a -3a b +3ab -b =(a -b )
(8)a +b +c +2ab +2bc +2ca =(a +b +c ) (9)a -b =(a -b )(a
n
n
n -1
2
2
2
2
3
2
2
3
3
3
2
2
3
3
3
3
2
2
+a
n -2
b + +ab
n -2
+b
n -1
), (n ≥2)
倒数关系:sinx·cscx=1 tanx·cotx=1 cosx·secx=1
商的关系:tanx=sinx/cosx cotx=cosx/sinx
平方关系:sin^2(x)+cos^2(x)=1 tan^2(x)+1=sec^2(x) cot^2(x)+1=csc^2(x)
倍角公式:
sin(2α)=2sinα·cosα
cos(2α)=cos^2(α)-s in^2(α)=2cos^2(α)-1=1-2sin^2(α) tan(2α)=2tanα/[1-tan^2(α)]
降幂公式:
sin^2(α/2)=(1-cosα)/2 cos^2(α/2)=(1+cosα)/2 tan^2(α/2)=(1-cosα)/(1+cosα) tan(α/2)=sinα/(1+cosα)=(1-cosα)/sinα
两角和差:
sin(α±β)=sinα·cosβ±cosα·sinβ cos(α+β)=cosα·cosβ-sinα·sinβ cos(α-β)=cosα·cosβ+sinα·sinβ tan(α+β)=(tanα+tanβ)/(1-tanα·tanβ) tan(α-β)=(tanα-tanβ)/(1+tanα·tanβ)
积化和差:
sinα·cosβ=(1/2)[sin(α+β)+sin(α-β)] cosα·sinβ=(1/2)[sin(α+β)-sin(α-β)] cosα·cosβ=(1/2)[cos(α+β)+cos(α-β)] sinα·sinβ=-(1/2)[cos(α+β)-cos(α-β)]
和差化积:
sinα+sinβ=2sin[(α+β)/2]cos[(α-β)/2] sinα-sinβ=2cos[(α+β)/2]sin[(α-β)/2] cosα+cosβ=2cos[(α+β)/2]cos[(α-β)/2] cosα-cosβ=-2sin[(α+β)/2]sin[(α-β)/2]
特殊角的三角函数值:
等价代换:
x x (4) a r c t a n ~x x (1) sinx ~x (2) tanx ~x (3) a r c s i n ~
x (5) 1-c o s ~
12x
2
a x
(6) ln (1+x ) ~x (7) e -1~x (8) (1+x ) -1~ax
基本求导公式:
(1) (C ) '=0 ,C 是常数 (2) (x α) '=αx α-1 (3) (a x ) '=a x ln a (4) (loga x ) '=
1x ln a
(5) (sinx ) '=cos x (6) (cosx ) '=-sin x
1cos
2
2
(7) (tanx ) '=
x
=sec
x (8) (cotx ) '=-
1sin
2
x
=-csc
2
x
(9) (s e x c ) '=(s e x c ) t a n x (10) (cscx ) '=-(cscx ) cot x (11) (arcsinx ) '=
1-x
2
(12) (arccosx ) '=-
1-x
11+x
2
2
(13) (arctanx ) '=
'=(15) (x )
12x
11+x
2
(14) (arccot x ) '=-
1x
1x
2
()=- (16)
基本积分公式: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)
⎰0dx =C ⎰kdx
=kx +C
(k 为常数)
+C
⎰x
1
μ
dx =
x
μ+1
μ+1
(μ
≠-1)
⎰x dx
=ln |x |+C
⎰a
x
dx =
a
x
ln a
x
+C
⎰e dx =e +C ⎰cos xdx ⎰sin ⎰cos
x
=sin x +C
xdx =-cos x +C
==
dx
2
x
2
⎰sec
2
xdx =tan x +C
2
⎰sin
dx x
⎰csc
xdx =-cot x +C
⎰sec x tan
xdx =sec x +C
dx 1+x
2
⎰csc x cot xdx ⎰1+x
dx
2
=-csc x +C
=arctan x +C
或(⎰
=-arc cot x +C
)
⎰
dx -x
2
=arcsin x +C 或(⎰
dx -x
2
=-arccos x +C )
⎰tan xdx =-ln |cos x |+C , ⎰cot xdx =ln |sin x |+C , ⎰sec xdx =ln |sec x +tan x |+C , ⎰c sc x dx
=ln |csc x -cot x |+C
,
高数常用公式
平方立方:
(1)a -b =(a +b )(a -b ) (2)a +2ab +b =(a +b ) (3)a -2ab +b =(a -b )
3
3
2
2
2
2
2
2
2
2
2
2
(4)a +b =(a +b )(a -ab +b ) (5)a -b =(a -b )(a +ab +b ) (6)a +3a b +3ab +b =(a +b ) (7)a -3a b +3ab -b =(a -b )
(8)a +b +c +2ab +2bc +2ca =(a +b +c ) (9)a -b =(a -b )(a
n
n
n -1
2
2
2
2
3
2
2
3
3
3
2
2
3
3
3
3
2
2
+a
n -2
b + +ab
n -2
+b
n -1
), (n ≥2)
倒数关系:sinx·cscx=1 tanx·cotx=1 cosx·secx=1
商的关系:tanx=sinx/cosx cotx=cosx/sinx
平方关系:sin^2(x)+cos^2(x)=1 tan^2(x)+1=sec^2(x) cot^2(x)+1=csc^2(x)
倍角公式:
sin(2α)=2sinα·cosα
cos(2α)=cos^2(α)-s in^2(α)=2cos^2(α)-1=1-2sin^2(α) tan(2α)=2tanα/[1-tan^2(α)]
降幂公式:
sin^2(α/2)=(1-cosα)/2 cos^2(α/2)=(1+cosα)/2 tan^2(α/2)=(1-cosα)/(1+cosα) tan(α/2)=sinα/(1+cosα)=(1-cosα)/sinα
两角和差:
sin(α±β)=sinα·cosβ±cosα·sinβ cos(α+β)=cosα·cosβ-sinα·sinβ cos(α-β)=cosα·cosβ+sinα·sinβ tan(α+β)=(tanα+tanβ)/(1-tanα·tanβ) tan(α-β)=(tanα-tanβ)/(1+tanα·tanβ)
积化和差:
sinα·cosβ=(1/2)[sin(α+β)+sin(α-β)] cosα·sinβ=(1/2)[sin(α+β)-sin(α-β)] cosα·cosβ=(1/2)[cos(α+β)+cos(α-β)] sinα·sinβ=-(1/2)[cos(α+β)-cos(α-β)]
和差化积:
sinα+sinβ=2sin[(α+β)/2]cos[(α-β)/2] sinα-sinβ=2cos[(α+β)/2]sin[(α-β)/2] cosα+cosβ=2cos[(α+β)/2]cos[(α-β)/2] cosα-cosβ=-2sin[(α+β)/2]sin[(α-β)/2]
特殊角的三角函数值:
等价代换:
x x (4) a r c t a n ~x x (1) sinx ~x (2) tanx ~x (3) a r c s i n ~
x (5) 1-c o s ~
12x
2
a x
(6) ln (1+x ) ~x (7) e -1~x (8) (1+x ) -1~ax
基本求导公式:
(1) (C ) '=0 ,C 是常数 (2) (x α) '=αx α-1 (3) (a x ) '=a x ln a (4) (loga x ) '=
1x ln a
(5) (sinx ) '=cos x (6) (cosx ) '=-sin x
1cos
2
2
(7) (tanx ) '=
x
=sec
x (8) (cotx ) '=-
1sin
2
x
=-csc
2
x
(9) (s e x c ) '=(s e x c ) t a n x (10) (cscx ) '=-(cscx ) cot x (11) (arcsinx ) '=
1-x
2
(12) (arccosx ) '=-
1-x
11+x
2
2
(13) (arctanx ) '=
'=(15) (x )
12x
11+x
2
(14) (arccot x ) '=-
1x
1x
2
()=- (16)
基本积分公式: (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)
⎰0dx =C ⎰kdx
=kx +C
(k 为常数)
+C
⎰x
1
μ
dx =
x
μ+1
μ+1
(μ
≠-1)
⎰x dx
=ln |x |+C
⎰a
x
dx =
a
x
ln a
x
+C
⎰e dx =e +C ⎰cos xdx ⎰sin ⎰cos
x
=sin x +C
xdx =-cos x +C
==
dx
2
x
2
⎰sec
2
xdx =tan x +C
2
⎰sin
dx x
⎰csc
xdx =-cot x +C
⎰sec x tan
xdx =sec x +C
dx 1+x
2
⎰csc x cot xdx ⎰1+x
dx
2
=-csc x +C
=arctan x +C
或(⎰
=-arc cot x +C
)
⎰
dx -x
2
=arcsin x +C 或(⎰
dx -x
2
=-arccos x +C )
⎰tan xdx =-ln |cos x |+C , ⎰cot xdx =ln |sin x |+C , ⎰sec xdx =ln |sec x +tan x |+C , ⎰c sc x dx
=ln |csc x -cot x |+C
,