integralen van
goniometrische functies |
primitieve functie en afgeleide
F(x) is een primitieve functie van de functie f(x) als
de afgeleide van F(x) = f(x)
als |
F ' (x) = f(x), |
dan |
 |
f(x) dx = F(x) + c. |
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rekenregels
afgeleide |
primitieve functie |
d
 dx |
sin x = cos x |
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 |
cos x dx = sin x + C |
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d
 dx |
cos x = sin x |
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 |
sin x dx = cos x + C |
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d
 dx |
tan x = sec2x |
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 |
sec2x dx = tan x + C |
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d
 dx |
cotan x = cosec2 x |
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 |
cosec2x dx = cotan x + C |
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d
 dx |
sec x = sec x tan x |
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 |
(sec x tan x) dx = sec x + C |
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d
 dx |
cosec x = cosec x cotan x |
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 |
(cosec x cotan x) dx = cosec x + C |
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Oefeningen
Bereken de volgende integralen
a) |
 |
(3sin x 4sec2x) dx |
b) |
 |
cos(2x 6) dx |
c) |
 |
sin x cos2x dx |
d) |
 |
tan x dx |
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Oplossingen
a) |
 |
(3sin x 4sec2x) dx |
 |
(3sin x 4sec2x) dx |
= |
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(splitsen in som van integralen) |
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= |
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(buiten haakjes zetten van constante) |
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= |
3cos x 4tan x + C
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(basisintegralen uit tabel) |
b) De integraal
cos(2x
6) dx
kunnen we oplossen door substitutie
u = 2x 6 |
du
 dx |
= |
2 |
dx |
= |
1
 2 |
du |
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Zo krijgen we
 |
cos(2x 6) dx |
= |
 |
cos u |
1
 2 |
du |
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(substitutie) |
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|
= |
1
 2 |
 |
cos u du |
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(buiten haakjes zetten van constante) |
|
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= |
1
 2 |
sin u + C |
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(basisintegralen uit tabel) |
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|
= |
| 1
 2 |
sin(2x 6) + C. |
|
(we vervangen u opnieuw) |
c) Ook de integraal
sin x cos2x
kunnen we oplossen door substitutie toe te passen
u = cos x |
du
 dx |
= sin x |
dx |
=  |
1
 sin x |
du |
|
We krijgen nu
 |
sin x cos2x dx |
= |
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(substitutie) |
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= |
 |
 |
u2 du |
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= |
 |
u3
 3 |
+ C |
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|
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= |
|  |
cos3x
 3 |
+ C. |
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(we vervangen u opnieuw) |
d) We herschrijven
tan x dx als
(sin x / cos x) dx,
en gebruiken dezelfde substitutie als in de vorige oefening:
 |
tan x dx |
= |
 |
sin x
 cos x |
dx |
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= |
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(substitutie) |
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= |
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= |
ln (u) + C
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= |
ln (cos x) + C. |
(we vervangen u opnieuw) |
uitbreiding van
formules
Uit de oefeningen met substitutie leren we:
Basisregel
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Uitbreiding
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cos x dx = sin x + c |
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cos(ax+b) dx = |
1
 a |
sin(ax+b) + c |
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sin x dx = cos x +
c |
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sin(ax+b) dx =  |
1
 a |
cos(ax+b) + c |
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tan x dx = ln (cos x)
+ c |
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tan(ax+b) dx =  |
1
 a |
ln (cos(ax+b))+ c |
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cotan x dx = ln (sin x) + c
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cotan(ax+b) dx = |
1
 a |
ln (sin(ax+b)) + c |
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sec x dx = ln (sec x + tan x) + c
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sec(ax+b) dx = |
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1
 a |
ln (sec(ax+b) + tan(ax+b))+ c |
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cosec x dx = |
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ln (cosec x + cotan x) +
c |
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cosec(ax+b) dx = |
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 |
1
 a |
ln (cosec(ax+b) + cotan(ax+b))+ c |
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