B - Arnout Devos

Partieelbreuken: oefening
6
8
x
x
dx
)
1
( 2
6 x
x
dx
: splitsen in partieelbreuken
1
)
1
(
1
2
6
5
4
3
2
2
6
x
H
Gx
x
F
x
E
x
D
x
C
x
B
x
A
x
x
6
7
2
3
2
4
3
5
4
6
5
7
1
Hx
Gx
F
Fx
Ex
Ex
Dx
Dx
Cx
Cx
Bx
Bx
Ax
Ax
1
2
6
4
2
6
8
x
dx
x
dx
x
dx
x
dx
x
x
dx
C
x
Bg
x
x
x
tan
5
1
3
1
1
5
3
1
F
Dan:
7
6
5
4
3
2
)
(
)
(
)
(
)
(
)
(
)
(
1
x
G
A
x
H
B
x
C
A
x
D
B
x
E
C
x
F
D
Ex
F
0
E 1
D 0
C 1
B 0
A 1
H 0
G

xdx
1 x
2
ln
y
cos 3
y
dy
xsin
1
2
1
ln
2
dt
t
xdx
t C
tdt
2 3/
2
t
3
C
Integralen
1 2
ln( 1
x ) C
2
t dt
3
2
3
4
t
C
4
(ln
3
y) C
1
4
cos
4
x C
Substitutie:
Substitutie:
Substitutie:
2 1
x t
2xdx
dt
ln
y t
dy
dt
y
cos
x t
sin xdx dt

Integralen
x
x
x
x
b
a
dx
b
a
2
)
(
dx
b
a
b
a
du
b
a
u
x
x
ln
Substituties:
dx
b
a
b
b
a
a
x
x
x
x
x
x
2
2
2
dx
a
b
dx
dx
b
a
x
x
x
x
2
dx
a
b
a
b
dv
a
b
v
x
x
ln
a
b
du
x
b
a
du
ln
2
ln
C
a
b
a
b
x
b
a
b
a
x
x
ln
ln
1
2
ln
ln
1
C
x
b
a
a
b
b
a
x
x
2
ln
ln
1
a
b
u
x
b
a
u
ln
2
ln

Integralen
dx
x
x
4
1
4
/
4
1
dt
dx
t
x
Substitutie:
dt
t
t
)
1
(
16
1
t
dt
dt
t
16
1
16
1
C
x
x
2
/
1
2
/
3
)
4
1
(
8
1
)
4
1
(
24
1
d
3
sin
d
sin
sin 2
cos
)
cos
1
(
2
d
dt
d
t
cos
cos
Substitutie:
dt
t
dt
2
C
3
cos
3
1
cos

Integralen (Oef. PI)
2
2 2x
ln( x 2)
dx x ln( x 2)
x dx
2
x 2
2
2 x 2 2
x ln( x 2)
2
dx
2
x 2
2
1
x ln( x 2)
2x
4
dx
2
x 2
2 4 1
x ln( x 2)
2x
dx 2
2 x / 2 1
2
x ln( x 2)
2x
2
dt
2
2
1
t
2
xln( x 2)
2x
2 2Bg
tan
x
C
2
2
2 x
x ln( x 2)
2
dx 2
x 2
Substitutie:
x
/ 2
dx
t
2dt

Integralen (Oef. PB ) (1/2)
?
)
5
2
)(
1
(
3
3
2
2
2
dx
x
x
x
x
x
Splitsen in partieelbreuken:
dx
x
x
x
x
dx
5
2
2
3
1
2
5
2
1
)
5
2
)(
1
(
3
3
2
2
2
2
x
x
C
Bx
x
A
x
x
x
x
x
C
Cx
Bx
Bx
A
Ax
Ax
x
x
2
2
2
5
2
3
3
2
C
A
x
C
B
A
x
B
A
5
)
2
(
)
(
2
Hieruit volgt:
3
5
3
2
2
C
A
C
B
A
B
A
2
3
1
C
B
A
dx
x
x
x
x
x
)
5
2
)(
1
(
3
3
2
2
2

Integralen (Oef. PB) (2/2)
dx
x
x
x
x
dx
5
2
2
3
1
2
dx
x
x
x
x
x
)
5
2
)(
1
(
3
3
2
2
2
dt
x
dx
t
x
x
1
2
)
2
2
(
5
2
Substitutie:
dx
x
x
x
x
dx
5
2
3
/
2
2
2
2
3
1
2
dx
x
x
dx
x
x
x
x
dx
5
2
1
5
2
2
2
2
3
1
2
2
dx
x
t
dt
x
4
)
1
(
1
2
3
)
1
ln( 2
dx
x
x
x
x
1
2
/
)
1
(
1
4
1
)
5
2
ln(
2
3
)
1
ln( 2
2
C
x
Bg
x
x
x
2
1
tan
2
1
)
5
2
ln(
2
3
)
1
ln(
2
du
dx
u
x
2
/
2
/
)
1
(
Substitutie: