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Alles voor Analyse 1 NA (samenvatting)
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Samenvatting met de volledige stof van Analyse 1 NA
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Calculus 2 Notes (Integral Calculus)
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Summary Calculus 2 (2WBB0) 2020/2021
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IntegrerenLijst met primitieven van standaardfuncties:
I .
Sxnax =
IT ,
t c Cnt D -
7 .
Jcoshcxs -
- sinhlxs
§cost
8. ,
dx = tanks to
2.) txdx In late g.
J¥÷
-
. Incisor = xlncxs -
xtc
s . S wax -
-
II ,
te Caso ,
a # D co .
) ax : antanas to
,
= tearoom (E) to →
4 .
I sink> ax : -
costs dx l l .
J ydx = are sink) t c
S = arcos Cx) to
-
5 .
cos ↳ dx= sin ↳ de
6 .
fsinh ↳ = cosh ax
Trigonometrische integralen
} sinmlx)
"
cos (x) dx M, h C- I en Muzo
We onderscheiden twee gevallen:
1. minstens 1 van de getallen m,n is oneven
2. m,n zijn beide even
Vb .
geralt :
J sink) ✓
oh even
cos %) dx I
f cost ( I -
cost x) ) sink) dx =
-
Sub ( i - u2 ) du =
-
Hutt Just c
sink) t cos 24=1 macht UI cos ( x) =
-
I cost tty cos %) to
'
[cos G) ) -
-
-
sin Cx)
du = -
sin G) dx
'
[ sin (D) = cos G)
vb .
S sin '
(x) cos 5lb dx alle bei on even
(i ) =
)( l -
cos
'
cos 54 sin dx =
-
f( I -
u )2u5du ( u
-
- cos
(ii) sink) du
u : ,
-_cos↳ cosh Cx)
{ Pcos
'
J )
'
sink x) ( l -
sin dx = u 'll -
w du
Ub .
gevul 2 : Jsinhlxscoshlxdr =
- gebruik sin^2(x)+cos^2(x)=1 om de integraal te schrijven als een som van integralen van even machten van cos(x)
=
)( I -
cosy# cos
'
Cx) -
-
f costly -2 cos GH t cos Rx) or sinks
"
Jcos4x)dx = SI t tacos (2x) = { x t t sin ( 2x) to Scoshksdx ) : ( t tf cost dr.ftttzwsczxsttc.im
cos (2x) - Leos' ↳ -
I = I -
2 sin Cx) = text 4 since t Tf cop dx : thx ttgsinczxttftzttzcoslhxlox
cosy = It { cos (2x) = text t sin last text 's sin ( Htc
sinks =
-
I -
{ cos Las
Integratiemethoden:
Substitutiemethode: Inverse substitutie:
du - n' ↳ dx X' (a) du=d×
, Een paar standaardsubstituties:
Als in de integraal een factor
•
U2 -
x
'
Voorkomt , pro beer clan x = a cos Cu) of x
-
- a sin (us
• ¥2 → x- ucoshcus
,
I -
u2 : a sinh ( u)
x -
-
a Sec ( us
,
E -
a
>
= atanlu
want Tanya) ti : Sedler)
• Eta → * usinhcu) ,
Etah = a cosh ( us
E- utuncu ,
Itu = useccu)
Partieel integreren:
SF 'HgHdx :
FLASH -
Spurs g '
Cx) dy
Rationele functies integreren:
Breuksplitslemma
PLA
grand (Psc graced ( Q)
-
—> dan kan je breuksplitsen (zie vb. p10 notities h5 of hw5)
Qlxs
6.4: Methode van onbepaalde coëfficiënten Werft niet voor alle soon ten Funches
-
Vb .
J x
'
e' ×
ax = ( axztbxt c) e
"
t E :( 5×2 -
Txt Idek to
:( 3×2 t 2x)
' ''
[x' e
't
] e
x' e" = ( Jux't 3bar t 3C t 2u× tb) e
"
Ju : I a :} ,
b= -
} .
⇐ 37
Tbt 2u= 0 → Ib : -
}
It b - O 3C :
}
Ub .
J e
"
cos thx) dx
;
e
"
( ucoslzx) t b sin (2x) ) t c
Ansatz
( e" cosh xD
'
= e3×( 3 cos GD -
25in (2x))
3 at 2b al
-
a
-
- E
→
↳ →
e. cost 2x> = e) * ( 3u cost 2x) t 2b sinus -
Zu sin ( 2x) t 2b cos Cls) Jb - 2u= O b -
-
}
i
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