1. POCHODNE FUNKCJI ELEMENTARNYCH

b

b

R

R

(C)

u(x)v0(x)dx = [u(x) · v(x)]b − u0(x)v(x)dx , przez 0=0

a

a

a

(xa)0 = axa−1

cz¦±ci

(sin x)0 = cos x

b

P(D)=R

(

[f (x) − g(x)]dx ,pole obrzaru normalnego cos x)0 = − sin x

a

(tg x) = 1

D

cos2 x

(ctg x)0 = − 1

V=π R ]limitsb[f(x)]2dx ,obj¦to±¢ bryªy obrotowej sin2 x

a

(ax)0 = ax ln a

b

q

R

(e

S=2

x

π

f (x)

1 + [f 0(x)]2 pole powierzchni bryªy ob-

)0 = ex

a

(log x)0 = 1

rotowej

a

x ln a

(

ln x)0 = 1

δf

δf

x

f(x

(x

(x

,gradient f. wielu

(

0, y0) =

0, y0)

0, y0)

arcsin x)0 =

1

√

δx

δy

1−x2

(arccos x)0 = − 1

zmiennych

√1−x2

δf

δf

(arctg x)0 = 1

(x0, y0) = 0 ∧

(x0, y0) = 0 ,warunek ko-

1+x2

δx

δy

(arcctg x)0 = − 1

nieczny istnienia ekstremum 1+x2

(sinh x)0 = cosh x

6. CAŠKA PODWÓJNA

(

"

#

cosh x)0 = sinh x

b

k(x)

RR

R

R

(tg x)0 =

1

f (x, y)dxdy =

f (x, y)dy dx

cosh2 x

a

(

D

h(x)

ctgh x)0 = −

1

sinh2 x

,zamiana na iterowan¡

d b

2. WZORY NA PODSTAWOWE CAŠKI RR f (x, y)dxdy = R R f (x, y)dx dy R

D

c

a

dx = x + C

,zamiana na iterowan¡

R xndx = 1 xn+1

δx

δx

n+1

R

1 dx = ln |x| + C

δu

δv

x

R axdx = ax + C

J(u,v)=

ln a

R exdx = ex + C

δy

δy

R

dx

√

= arcsin x + C

δu

δv

1−x2

s

2

2

R

−dx

δF

δF

√

= arccos x + C

1−x2

S=RR

1 +

+

dxdy ,pole pªata po-

R sin xdx = − cos x + C

δx

δy

D

R

wierzchni

cos xdx = sin x + C

R

dx

m=RR

= tg x + C

u(x, y)dxdy ,masa obrzaru pªaskiego cos2 x

D

R

dx

= − ctg x + C

sin2 x

My

Mx

R

dx

x

y

,±rodek ci¦»ko±ci

= arctg x + C

s =

s =

1+x2

m

m

R

−dx = arcctg x + C

My = RR xu(x, y)dxdy

Mx = RR yu(x, y)dxdy

1+x2

D

D

I

y2µ(x, y)dxdy

I

(x2 + y2)µ(x, y)dxdy

3. WZORY TRYGONOMETRYCZNE

x = RR

0 = RR

D

D

sinh2 x + cosh2 x = 1

Iy = RR x2µ(x, y)dxdy , moment bezwªadno±ci sinh(2x) = 2 sinh x cosh x

D

7. CAŠKA POTRÓJNA

cosh(2x) = cosh2 x + sinh2 x

√

" ψ(x,y)

#

arcsinh x = ln(x +

x2 + 1)

RRR

R

√

f (x, y, z)dxdydz = RR

f (x, y)dz dxdy

arccosh x = ln(x +

x2 − 1)

Ω

D

ϕ(x,y)

(

"

#

)

arctgh x = 1 ln( x+1 )

b

h(x)

ψ(x,y)

2

1−x

=RR

R

R

f (x, y, z)dz dy

dx

arcctgh x = 1 ln( x+1 )

2

1−x

a

4. PODSTAWIENIA UNIWERALNE

g(x)

ϕ(x,y)

,zamiana na iterowan¡

t=tg(x)

RRR

2

f (x, y, z)dxdydz =

x=2arctg t

Ω

RRR

dx = 2dt

f [x(u, v, w), y(u, v, w), z(u, v, w)]|J (u, v, w)|dudvdw 1+t2

Ω

sin x = 2t

1+t2

, J(u, v, w) 6= 0 , zamiana zmiennych cos x = 1−t2



1+t2

x = %cosϕ = γsinψcosϕ

5. CAŠKA OZNACZONA





y = %sinϕ = γsinϕ

b

g(b)

R f (g(x)) g0(x)dx = R f (t)dt , przez podstawienie



z = γcosψ

a

g(a)

1

8. PODSTAWIENIA EULERA

√

R R(x, ax2 + bx + c)dx

√

√

1oa > 0 ax2 + bx + c = t ± ax

√

√

2oc > 0 ax2 + bx + c = tx ± c

√

3o4 > 0 ax2 + bx + c = t(x − xo) 2