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. CAKA 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 CAKI 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. CAKA 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. CAKA 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
√
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