ap aq = ap+q z = a + ib = r(cos ✓ + i sin ✓) = rei✓ ,

ap /aq = ap q r2 = a2 + b2 , tan ✓ = b/a (a 6= 0)

a q = 1/aq
(ap )q = ap·q f (x + h) f (x)
p f 0 (x) = lim
a1/p = pa h!0 h
a p bp = (ab)p
ap /bp = (a/b)p (u + v)0 = u0 + v 0
(c u(x))0 = c u0 (x) c
ln(ab) = ln a + ln b (u · v)0 = u0 v + uv 0
u0 v uv 0
ln(a/b) = ln a ln b (u/v)0 =
v2
ln(ap ) = p · ln a ✓ ◆
df df du
loga (b) = ln(b)/ ln(a) (f (u))0 = f 0 (u) · u0 = ·
dx du dx

u u sin u cos u tan u (xr )0 = rxr 1

0 0 0 p1 0p (ex )0 = ex
⇡/6 30 p1/2 p 3/2 1/ 3 (ax )0 = ax ln a (a > 0)
⇡/4 45 2/2 2/2 1
p p (ln x)0 = 1/x
⇡/3 60 3/2 1/2 3
⇡/2 90 1 0 (loga x)0 = 1/(x ln a)
(sin x)0 = cos x
(cos x)0 = sin x
1 = sin2 u + cos2 u (tan x)0 = 1/ cos2 x = 1 + tan2 x
p
tan u = sin u/ cos u (arcsin x)0 = 1/ 1 x2 ,
p
sin u = sin(u + 2⇡n), n 2 Z (arccos x)0 = 1/ 1 x2 ,
cos u = cos(u + 2⇡n), n 2 Z (arctan x)0 = 1/(1 + x2 )
tan u = tan(u + ⇡n), n 2 Z
x0
sin(u) = sin(⇡ u)
f (x) ⇡ f (x0 ) + f 0 (x0 )(x x0 )
cos(u) = cos( u)
sin(u) = sin( u)
f (xn )
cos2 u = (1 + cos(2u))/2 xn+1 = xn
f 0 (xn )
sin2 u = (1 cos(2u))/2
sin(u + v) = sin u cos v + cos u sin v
cos(u + v) = cos u cos v sin u sin v f (x) g(x)
1 1 x!a
sin(2u) = 2 sin u cos u
f (x) f 0 (x)
cos(2u) = cos2 u sin2 u lim = lim
x!a g(x) x!a g 0 (x)

y y0 = a(x x0 ) f 0 (x) > 0 x 2 (a, b) f

[a, b] f 0 (x) < 0 x 2 (a, b)
(x x0 )2 + (y y0 ) 2 = r 2 f [a, b]
 Z b
n
X Z b
f (x) dx ⇡ Tn
f (xi ) xi ! f (x) dx a ⇣ ⌘
a
i=1
Tn = x f (x2 0 ) + f (x1 ) + · · · + f (xn 1) + f (xn )
2
xi ! 0 i
x = (b a)/n xi = a + i x

Z Z b
b
0
F (x) dx = F (b) F (a) f (x) dx ⇡ Sn n =
a
a
Z b Z b Sn = 3x (f (x0 ) + 4f (x1 ) + 2f (x2 ) + 4f (x3 ) +
cf (x) dx = c f (x) dx · · · + 2f (xn 2 ) + 4f (xn 1 ) + f (xn )),
a a x = (b a)/n xi = a + i x
Z b Z b Z b
f (x)
(f + g) dx = f dx + g dx
a a a Rbp
Z b Z b L= 1 + (f 0 (x))2 dx
a
uv 0 dx = [uv]ba u0 v dx x
a a Rb
Z b Z u(b)
Vx = ⇡ a f (x)2 dx
f (u)u0 dx = f (u) du y
Rb
a u(a) Vy = 2⇡ a xf (x) dx

Z
xr+1 Z
xr dx = + C r 6= 1 1 b
r+1 y= y(x) dx
Z b a a
1
dx = ln |x| + C
x
Z
ex dx = ex + C Z Z
Z h(y) · y0 = g(x) h(y) dy = g(x) dx
sin x dx = cos x + C
Z ⇣ ⌘0
cos x dx = sin x + C y 0 + f (x)y = g(x) yeF (x) = g(x)eF (x)
Z ay 00 + by 0 + cy = 0
1 a 2+b +c=0
dx = tan x + C
cos2 x 1x 2x
Z y = Ae + Be ( )
1
p dx = arcsin x + C y = e (A + Bx) ( )x
1 x2
Z ↵x
y = e (A cos( x) + B sin( x))
1
dx = arctan x + C
1 + x2 =↵± i
ay 00 +by 0 +cy = g(x)
Z Z y = y h + yp yh
uv 0 dx = uv u0 v dx yp
Z Z g
0
f (u)u dx = f (u) du A
Z
1 det A 6= 0 A
f (ax + b) dx = F (ax + b) + C
a A~x = ~b
A
2⇥2
>
a b
= ad bc
(ax + b) A c d
ax+b
✓ ◆ 1 ✓ ◆
(ax + b)k a b 1 d b
A1 A2 Ak =
ax+b + (ax+b)2 + · · · + (ax+b)k c d ad bc c a
Ax+B
x2 + b2 x2 +b2
Derivasjonsregler Integrasjonsregler
R R R
! "0
af ðxÞ þ bgðxÞ ¼ af 0 ðxÞ þ bg 0 ðxÞ af ðxÞ þ bgðxÞ d x ¼ a f ðxÞ d x þ b gðxÞ d x
R R
f ðgðxÞÞ 0 ¼ f 0 ðgðxÞÞg 0 ðxÞ f ðgðxÞÞg 0 ðxÞ d x ¼ f ðuÞ d u
! " R R
f ðxÞgðxÞ 0 ¼ f 0 ðxÞgðxÞ þ f ðxÞg 0 ðxÞ f ðxÞg 0 ðxÞ d x ¼ f ðxÞgðxÞ % f 0 ðxÞgðxÞ d x
# $ Zx
f ðxÞ 0 f 0 ðxÞgðxÞ % f ðxÞg 0 ðxÞ d
¼ f ðtÞ d t ¼ f ðxÞ
gðxÞ ðgðxÞÞ2 dx
a

Deriverte av grunnfunksjonene Rb
F 0 ðxÞ d x ¼ FðbÞ % FðaÞ
r 0 r%1 1 a
ðx Þ ¼ rx ðln xÞ 0 ¼
x
ðex Þ 0 ¼ ex ðsin xÞ 0 ¼ cos x

ðcos xÞ 0 ¼ % sin x ðtan xÞ 0 ¼ 1 þ tan 2 x ¼

1 Tabell
Z
over nyttige integraler
cos 2 x 1 rþ1
xr d x ¼ x þ C, r 6¼ %1
x 0 x 0 loga e rþ1
ða Þ ¼ a & ln a ðloga xÞ ¼ Z
x 1
1 1 d x ¼ ln jxj þ C
0
ðarcsin xÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 0
ðarctan xÞ ¼ x
1 % x2 1 þ x2 Z
1
cos ax d x ¼ sin ax þ C
a
Trigonometriske sammenhenger Z
1
sin ax d x ¼ % cos ax þ C
sin 2 x þ cos 2 x ¼ 1 a
Z
sin x ax 1 ax
tan x ¼ e dx ¼ e þ C
cos x a
Z
sin ð%xÞ ¼ % sin x ln x d x ¼ x ln x % x þ C
cos ð%xÞ ¼ cos x Z
&! ' 1 1 x
d x ¼ arctan þ C
sin % x ¼ cos x a2 þ x2 a a
2 Z
&! ' 1 x
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi d x ¼ arcsin þ C
cos % x ¼ sin x a %x 2 2 a
2 Z
1 1 ((x þ a((
sin ð2! þ xÞ ¼ sin x d x ¼ ln( (þC
a2 % x2 2a x % a
cos ð2! þ xÞ ¼ cos x Z x
a
ax d x ¼ þC
sin ð! % xÞ ¼ sin x ln a
Z
x 1
cos ð! % xÞ ¼ % cos x sin 2 x d x ¼ % sin 2x þ C
2 4
Z
tan ð%xÞ ¼ % tan x x 1
cos 2 x d x ¼ þ sin 2x þ C
tan ð! þ xÞ ¼ tan x 2 4
Z
cos ax x sin ax
sin ðx þ yÞ ¼ cos x sin y þ sin x cos y x cos ax d x ¼ þ þC
a2 a
Z
cos ðx þ yÞ ¼ cos x cos y % sin x sin y sin ax x cos ax
x sin ax d x ¼ % þC
cos 2x ¼ cos 2 x % sin 2 x a2 a
Z
2 sin ax 2x cos ax x2 sin ax
cos 2x ¼ 1 % 2 sin 2 x x2 cos ax d x ¼ % þ þ þC
a3 a2 a
cos 2x ¼ 2 cos 2 x % 1 Z
2 cos ax 2x sin ax x2 cos ax
x2 sin ax d x ¼ þ % þC
sin 2x ¼ 2 sin x cos x a3 a2 a
1 þ cos 2x
cos 2 x ¼
2
2 1 % cos 2x
sin x ¼
2
1! "
sin mx cos nx ¼ sin ðm þ nÞx þ sin ðm % nÞx
2
1! "
cos mx cos nx ¼ cos ðm þ nÞx þ cos ðm % nÞx
2
1! "
sin mx sin nx ¼ cos ðm % nÞx % cos ðm þ nÞx
2
Noen rekker Regneregler for Laplace-transformasjonen
X
1 n
x
ex ¼ 1. f 0 ðtÞ sFðsÞ % f ð0Þ
n¼0
n!
2. f 00 ðtÞ s2 FðsÞ % sf ð0Þ % f 0 ð0Þ
X
1
n%1 xn
lnð1 þ xÞ ¼ ð%1Þ , x 2 h%1, 1' sn FðsÞ % sn % 1 f ð0Þ % sn % 2 f 0 ð0Þ % . . .
n¼1
n 3. f ðnÞ ðtÞ
% sf ðn%2Þ ð0Þ % f ðn%1Þ ð0Þ
X
1
x2n n
cos x ¼ ð%1Þ Rt
ð2nÞ! FðsÞ
n¼0 4. f ð"Þ d"
0 s
X
1
x2nþ1
sin x ¼ ð%1Þn 5. eat f ðtÞ Fðs % aÞ
n¼0
ð2n þ 1Þ!
X
m 6. uðt % cÞf ðt % cÞ e%cs FðsÞ
n 1 % xm þ 1
ax ¼ a
n¼0
1%x 1 &s'
7. f ðctÞ F
X
1 c c
a
axn ¼ , x 2 h%1, 1i
1%x ðf ) gÞðtÞ t
n¼0
8. R FðsÞ & GðsÞ
¼ f ð"Þgðt % "Þ d"
0

9. tf ðtÞ %F 0 ðsÞ
Fourier-rekker
For f definert for x 2 ½%!, !' er f ðtÞ R1
P
1 10. Fð#Þ d#
Ff ðxÞ ¼ a0 þ ðan cos nx þ bn sin nxÞ der t s
n¼1
Z!
1
a0 ¼ f ðxÞ d x
2!
%! Basisformler for Laplace-transformasjonen
Z! ! " ! "
1 f ðtÞ ¼ L%1 FðsÞ FðsÞ ¼ L f ðtÞ ,
an ¼ f ðxÞ cos nx d x
! 1
%! 1. 1 , s>0
Z! s
1
bn ¼ f ðxÞ sin nx d x 1
! 2. t , s>0
%! s2
For en periodisk funksjon f med periode T er
1 # $ n!
X 2!nx 2!nx 3. tn , s>0
Ff ðxÞ ¼ a0 þ an cos þ bn sin der sn þ 1
n¼1
T T
!ðp þ 1Þ
cZþ T 4. tp , s>0
1 sp þ 1
a0 ¼ f ðxÞ d x
T 1
c 5. eat , s>a
s%a
cZþ T
2 2!nx
an ¼ f ðxÞ cos dx b
T T 6. sin bt , s>0
c s2 þ b2
cZþ T
2 2!nx s
7. cos bt , s>0
bn ¼ f ðxÞ sin dx s2 þ b2
T T
c
b
For en jevn funksjon g med periode T ¼ 2L er 8. eat sin bt , s>a
ðs % aÞ2 þ b2
X1
!nx
FgðxÞ ¼ a0 þ an cos der s%a
n¼1
L 9. eat cos bt , s>a
ðs % aÞ2 þ b2
ZL ZL
1 2 !nx 1
a0 ¼ f ðxÞ d x an ¼ f ðxÞ cos dx 10. teat , s>a
L L L ðs % aÞ2
0 0
n!
For en odde funksjon h med periode T ¼ 2L er 11. tn eat , s>a
X1 ðs % aÞn þ 1
!nx
FhðxÞ ¼ bn sin der
L e%cs
n¼1 12. uðt % cÞ , s>0
s
ZL
2 !nx 13. $ðtÞ 1, s > 0
bn ¼ f ðxÞ sin dx
L L
%cs
0 14. $ðt % cÞ e , s>0