I Podstawowe

twierdzenia

(oznaczenia: a,b,c,C,k,p,q-

stale; n EN;

f,g,F

- funkcje)

f(x+h)-

f(x)

ff

= F + C <=> F' = f (dalej stal<t C opuszczamy)

f':

D, 3XH

f'(x)=lim

f

IHO

h

ff tw:-L [F (x) J:= F (b)- F (a) II

1) a) (cof)'

=c-f';

b) (J +g)' = f' +g'

1) a) fcof=Co

ff

; b)

f(J±g)=

ff±

fg

b

b

b

b

b

fcof=c,

ff

;

f(J±g)=

ff±

fg

II

II

II

II

II

b

b

2)

(Jog)'

=f'.g+

fog'

l:-

2)c.p.cz.:

ff'·g=fog-

ffog'

;

f f' . g = [f 0 g f fog'

II

II

( f)'

g:o f' 0 g - f . g'

(~)' g*O g'

3)

[>

=

g

g2

g2

4) c.p.p.:

4)

(go f)'

= (g' 0 f)'

f'

1) fg(f(x))of'(x)dx

= II 1 ( x

'() ) = u

II = fg(u)du

I lI~f(x)

1 x dx=du

2) fg(x)dx

= 11:==II~(~i)du 11= fg(f(

u ))0 f'( u )du III=r1(x) I(x)=u

b

fJ

1) fg(f(x)) 0

f'(x)dx

=

,()

_

: x a

I

1

b

= fg(u)du

1 x dy-duo

II

I

1/3

a

U

a I

fJ

X = 1(11)

b

2)fg(x)dx=

,

.xlal/3

= f g ( f ( u )) 0 f' (u ) du dx = 1 (lI)dll ,

I I

a

U

a I b

II

[>

p:= ax+b= u,a*

j

0: \fi=F

+c => fi(ax+b)

= ~F(ax+b)

+c

a

P

I

cl

:= f:

If f' = lnlfl+c

0

ffk 0 f'

k:' _1

fk+l +

2

f

'

k+l

5) (r1)'-

1

- f'or'

II Uzyteczne

wzory

Tl:

sin ax ocosbx = 0,5 ~Sin( a+b)x

+sin( a- b)X~

WI k = e

I;

blnll

W2

log b = loge b

[>

cosaxocosbx

= 0,5

cos( a + b)x + cos( a-b)x II

loge a

sinax-sinbx

=0,5

cos(a-b)x-cos(a+b)x

T2 : 1+ cos x = 2 cos2 ~

T-H

T3: sin

2

2 x = ~-~cos2x

I> sin~ x =!( cos4x-4cos2x+3) 228

sin2x+ cos2 x = 1

cosh2 X - sinh2 X = 1

2sinxcosx = sin2x

2sinhxcoshx= sinh2x

l-cosx

= 2sin2 ~

2

1

1

I>

~

l(

)

cos x =-+-cos2x

cos x =-

cos4x+4cos2x+3

cos2 x-sin2 x = cos2x

cosh2 x +sinh2 X = cosh 2x 2

228

cos-2 x=1+tg2x

cosh-2 x=1-tgh2x

I Podstawowe

twierdzenia

(oznaczenia: a,b,c,C,k,p,q-

stale; n EN;

f,g,F

- funkcje)

f(x+h)-

f(x)

ff = F + C <=> F' = f (dalej sta1tt C opuszczamy) 1': D, 3XH

I'(x)=lim

f

11-.+0

h

If

tw:-L [F(x)J: = F(b)-F(a) II

1) a) (cof)'

=c-I';

b) (J +g)' = I' +g'

1) a) fcof=Co

ff

; b) f(J±g)=

ff±

fg

b

b

b

b

b

fcof=Co

ff

;

f(J±g)=

ff±

fg

II

II

II

II

II

b

b

2)

(J. g)' = f' - g + fog'

2)c.p.cz.:

fl'·g=fog-

ffog'

;

f I' g = [f g! - f f . g'

0

0

II

II

( f)'

g:o I' g - f . g'

0

(~)'g:;t::O g'

3)

[>

=

g

g2

g2

4)

(go f)'

= (g' 0 f)' I'

4) c.p.p.:

1) fg(J(x))ol'(x)dx

= II f ( x

'() ) = u

II = f g(u)du

I lI~f(x)

f

x dx=du

2) fg(x)dx

= 11:=:f~ii)du

II = fg(J(

u )). f'( u )du III=r1(x)

f(x)=u

b

fJ

1) fg(J(x)).

I'(x)dx

=

,()

_

. x I a 1 b

= fg(u)du

II

f

XdY-du'ulaifJ

a

fJ

X = f(u)

b

2)fg(x)dx=

,

.xlalfJ

= fg(J(u))o

I'(u) du

dx = f (u)du ,

I I

a

U

a I b

II

[>

PI:= ax+b= u,a"* 0: Ifi = F+C => fi(ax+b)

= ~F(ax+b)

+c

a

P := f:

If I' = lnlfl+c

I ffk .I' k:1 _1 fk+l + cl 0

2

f

'

k+l

5) (r1)'-

1

- I'of-I

II Uzyteczne

wzory

Tl:

sinax ·cosbx

= 0,5 ~Sin( a+b)x

+sin( a- b)X~

cosa x·cosbx

= 0,5

cos( a + b)x+ cos( a-b)x sinax-sinbx

=0,5

cos(a-b)x-cos(a+b)x

T2 : 1+ cos x = 2 cos2 ~

T-H

T3: sin

2

2 x = ~-~cos2x

I> sin~ x =.!.( cos4x - 4cos2x + 3) 228

sin2x+cos2x=1

cosh2 X -sinh2 X = 1

2sinxcosx = sin2x

2sinhxcoshx= sinh2x

l-cosx

= 2sin2 ~

2

1

1

I>

~

l(

)

cos x =-+-cos2x

cos x =-

cos4x+4cos2x+3

cos2 x-sin2

x = cos2x

cosh2 x +sinh2 X = cosh 2x 2

228

cos-2 x=1+tg2x

cosh-2 x = I-tgh2x

!( Xk)' = k· Xk-lll> (c)' = 0; (x)' = 1 ; (X2)' = 2x;

!b=1 =>

fxk = k~1 Xk+11 v f

f

f

l'f,I,

(~)'

=-~.

(~)'

=-~.

0= C·

1= x·

x = -x-'

x- = -x '

,

,

2'

3'

2'

2

3

'

X

X

X

X

/12:::2

n

11/

(-£)'

f~=-~·

f~=

__ 1

. f_l

=2 ~.

f"~

= -x!\tx

= 2 I,X ; (if;)'

,

,

3

2 2'

,

'\IX,

'\IX

"~2

1

r

x

x

x

'\IX

n+l

"\IX

n!Jx"-1

f 1

1

1

n??2=>

(

)" =--1'(

)"-1

x-c

n-

x-c

I f~= 1n1xll '

,

W2

1

1

(loga x)

=

_.-

*

Ina

x

fin x dx c:;:z. f( x)' Inx dx = x(lnx-l)+

C

I fex = ex! v

WI

1

f

(

)'

WI

(

)(11)

aX

= Ina-ax;

eX

= eX

X

x

a = -·a

Ina

r

([f(x)]

:1

g(x)

[f(x)]g(x)

( g'(x).lnf(x)+

f'(;(~(x))

I

'

1

WYMIERNE)

(arctgx)

= -2-

f 1

X

+1

('>0

1

x

,

,

=

-arctg-

v

x- + c-

I(

C

C

arcctgx)'

x/+

=-

1-1

2

f

1

2

2x+ p

11 = P - 4q < 0 =>,

=

,---:- arctg

,---:-;

x- + px+q

'\1-11

'\1-11

fax

+ b

a

('

)

a p - 2 b

2x + P

,

= -In x- + p x + q

,---:- arctg

,---:-;

x- + px+q

2

'\1-11

'\1-11

Vn??2I:=f

1

=

,

11

(x2 + px+q)"

= _

1

[

2x + P

+ (4n _ 6) I

] v

( -1) 11 ( 2

)"-1

11-1

n

x + px+q

f

1

('>0

1

x

1

x

=

-·--+-arctg-

( 2

2 )2

2 c2 x2 + c2

2c3

C

X

+c

* farctgxdx c~cz. J(x)'arctgxdx = xarctgx-O,51n(x2 +1)+C

,

XE(-I,I)

1

( arcsin x)

=.J

'

1

00

x

1-x-

f 121 = arcsin - v

vc- -x-

c

,

XE(-I,I)

1

( arcccos x)

=

r:-----?

f

1

1.

-2ax-b

a<O =>

vl-x2

= -

arcsm

.Jax2 +bx+c

~

.Jb2 -4ac

~

1

~

c2

•

x_I

fvc-

-x-

dx = -xvc-

-x-

+-arcsm-

2

2

c

* farcsinxdx c:;:z. f(x)'arcsinxdx = xarcsinx+.Jl-x2

+C

r

1

J ri- ,:0 lnl

In( x + .J

x+.Jx2 +cl

V

X2

+ 1 = (arsinh x)'

.Jx2 + 1

\!X- + c

~lnl

a> 0 => J I 2 1

=

ax+~+~~ax2

+bx+c I

\lax

+bx+c

"a

2

og6lnie:

a ", 0 => J~ a x2 + b X + c =

=(~x+~

l~ax2 +bx+c _ b2-4ac

J

1

tt

2

4a)

8a

~ax2 +bx+c

(sin x)' = cos x

(sinhx)' = cosh x

JSinx = -cosx

JSinhx = cosh x

(cosx)' = -sinx

(cosh x)' = sinh

Jcosx = sin x

Jcoshx = sinh x

X

J

,

1

,

1

1

(tgx) =-2-

(tghx)

=

J-1,-=tgX

=tghx

cos- X

cosh

COS

X

cosh

2 x

2 X

,

1

,

1

(ctgx) =--.-

(ctghx) =--.-,-

J-.-1-

= -ctg X

J-.-1_,- = - ctgh x

sm

sm

slnh- x

2x

sinh- X

2x

I Jtgx = -lnlcosxl;

Jctgx = InIsin xl I sm x

2

cos x

2

4 II

J-.1

= InItg ~ I; J_1

= InI tg(~ + IT )

IJ .,

1

1.

2

J

2

1

1.

2 cl

sm-x=-x--sm

x·

cos x=-2x+-4sm

x

2

4

'

a",Oi

b",O =>

Jeaxsinbx=

a2~b2 (asinbx-bcosbx)eax;

J eax cosbx = _,_1__, (acosbx+ bsinbx)eax a- +b-k",O =>

JSinkx=-i

coskx;

Jcoskx=i

sinkx;

J x sinkx= J,sinkx- ~xcoskx;

k-

k

Jx coskx= ~coskx+

~xsinkx;

k

k

J2 • k

2

. k

1,

2

X sm x = -2 xsm

x- -x- cosk x+-cosk

x;

k

k

k3

J'

k

2

k

12,

2.

x- cos x = -2 XCOS x+ -x smkx--smkx

k

k

e

" -

" -

!(n-1)!!

n"

2

2

JSin" x dx: = Jcos"x dx: =

..

o

0

(n-1)!!.lT

n!!

2

a

f ci&gla i nieparzysta w (-a,a) ~ f f(x)dx

= 0;

o

a

f ci&gla i parzysta w (-a,a) ~ f f(x)dx

=2 ff(x)dx

-0

0