W10

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k=0
a
k
x
k
q(x) =
k=0
b
k
x
k
n < m
n
A
(ax + b)
k
Bx + C
(cx
2
+ dx + e)
l
,
m
A
B
C
a
b
c
d
e
d
2
− 4ce < 0
k
l
1
T
WIERDZENIE
ax + b
dx =
A
a
ln |ax + b| + C,
a
= 0
A
(ax + b)
n
dx = −
A
a(n − 1)(ax + b)
n−1
+ C, a
= 0, n
∈
N
\ {1}
A
ax + b
dx =
t = ax + b
dt = a dx
=
A
a
dt
t
=
A
a
ln |t|+C =
A
a
ln |ax+b|+C
A
(ax + b)
n
dx =
t = ax + b
dt = a dx
=
A
a
t
−n
dt =
A
a
t
−n+1
−n + 1
+ C
= −
A
a(n − 1)(ax + b)
n−1
+ C
q
n
m
A
2
T
WIERDZENIE
dx
x
2
+ a
2
a
arctg
x
=
a
+ C, a > 0
dx
(x
2
+ 1)
n
=
2(n − 1)(x
2
+ 1)
n−1
+
2n − 3
x
dx
(x
2
+ 1)
n−1
, n
∈
N
\ {1}
2n − 2
3 Przykład
dx
x
2
− 2x + 2
x
2
− 2x + 2 = (x − 1)
2
+ 1.
−4
dx
x
2
− 2x + 2
=
dx
(x − 1)
2
+ 1
=
t = x − 1
dt = dx
=
dt
t
2
+ 1
= arctg
t
1
+ C = arctg (x − 1) + C.
dx
(x
2
+ 2x + 5)
3
=
dx
((x + 1)
2
+ 4)
3
=
dx
=
dx
3
2
3
4
(x+1)
2
4
+ 1
4
3
x+1
2
+ 1
2
3
4
t =
x + 1
=
2
dt =
dx
2
dx = 2dt
5
(n=3)
=
1
2
6
2dt
(t
2
+ 1)
3
=
1
2
5
2 2(t
2
+ 1)
2
+
3
t
dt
(t
2
+ 1)
2
4
(n=2)
=
t
2
7
(t
2
+ 1)
2
+
3
2
7
2 1(t
2
+ 1)
+
1
t
dt
t
2
+ 1
2
=
t
2
7
(t
2
+ 1)
2
+
3t
2
8
(t
2
+ 1)
+
3
2
8
arctg t + C
=
x+1
2
2
+
3
x+1
2
+
3
2
8
arctg
x + 1
2
+ C
2
x+1
2
2
2
7
x+1
2
+ 1
2
8
+ 1
=
x + 1
2
4
(x
2
+ 2x + 5)
2
+
3(x + 1)
2
7
(x
2
+ 2x + 5)
+
3
2
8
arctg
x + 1
2
+ C
(x
2
+ px + q)
n
=
P
P x + Q
2x + p
(x
2
+ px + q)
n
+
Q −
P p
2
1
(x
2
+ px + q)
n
2
t = x
2
+ px + q
1
4 Przykład
p/q
p
q
x = t
N
,
x
5 Przykład
N
dx
dx
x
1/2
+ x
1/3
x = t
6
dx = 6t
5
dt
6t
5
dt
t
3
+ t
2
p/q
t
3
dt
t + 1
√
√
x
=
=
=
= 6
x +
3
t
2
− t + 1 −
1
t + 1
t
3
3
−
t
2
2
= 6
dt = 6
+ t − ln(t + 1)
+ C
= 2t
3
− 3t
2
+ 6t − 6 ln(t + 1) + C
= 2
√
√
√
√
x − 3
3
x + 6
6
x − 6 ln(
6
x + 1) + C, x > 0
x
ax + b
p/q
ax + b = t
N
,
p
q
6 Przykład
N
4
3x − 7 = t
4
3
p/q
√
dx = 4t
3
dt
dx =
4
5
t
4
3
t
3
dt =
4
t
5
5
4
3x − 7 dx =
=
+ C
3
3
t
3
dt
4
15
(3x − 7)
5/4
+ C,
≥
7
3
=
x
2
x − 5 = t
2
dx = 2t dt
x = t
2
+ 5
3
x
√
x − 5 dx =
4
5
=
(t
2
+ 5) t 2t dt
= 2
(t
4
+ 5t
2
) dt =
2t
5
5
+
10t
3
3
+ C = 2
t
4
5
+
5t
2
3
t + C
= 2
(x − 5)
2
5
+
5(x − 5)
3
√
x − 5 + C = 2
x
2
5
−
x
3
−
10
3
+ C
2
ax + b
cx + d
,
ad − bc
x
= 0,
p/q
ax + b
cx + d
= t
N
,
p
q
N
p/q
2
x + 1
x − 1
= t
3
x =
t
3
+ 1
t
3
− 1
3
3
x + 1
x − 1
dx
x + 1
=
4
5
=
t
t
3
−1
+ 1
dt
−6t
2
dt
(t
3
− 1)
2
dx =
= −6
t
(t
3
− 1)(t
3
+ 1 + t
3
− 1)
dt = −3
dt
t
2
(t
3
− 1)
=
= −3
−
dt
t
2
+
1
3
dt
t − 1
−
1
3
(t + 1) dt
t
2
+ t + 1
=
−3
t
− ln |t − 1| +
1
2
(2t + 1) dt
t
2
+ t + 1
+
1
2
dt
(t +
2
)
2
+
4
−3
t
− ln |t − 1| +
1
√
3
arctg
2t + 1
=
2
ln |t
2
+ t + 1| +
√
3
+ C,
t =
3
x + 1
x − 1
8
T
WIERDZENIE
dx
|a|
+ C,
√
= arcsin
a
2
− x
2
√
x
2
+ k
= ln
x +
x
2
+ k
+ C,
k
= 0, x
2
+ k > 0,
a
2
− x
2
dx =
x
2
a
2
− x
2
+
a
2
2
arcsin
|a|
+ C,
7 Przykład
−6t
2
(t
3
−1)
2
t
3
+1
dx
√
x
2
dx
= −
x
2
a
2
− x
2
+
a
2
2
arcsin
|a|
+ C,
a
2
− x
2
x
2
+ k dx =
x
2
x
2
+ k +
k
2
ln
x +
x
2
+ k
+ C,
k
= 0, x
2
+ k
≥ 0,
x
2
dx
√
x
2
+ k
=
x
2
x
2
+ k −
k
2
ln
x +
x
2
+ k
+ C,
k
= 0, x
2
+ k > 0.
9 Przykład
dx
dx
x + 1 =
√
5 t
√
5 dt
√
=
=
√
=
√
4 − 2x − x
2
5 − (x + 1)
2
dx =
5 dt
5 − 5t
2
=
√
dt
= arcsin t + C = arcsin
x + 1
√
+ C,
|x + 1|
<
√
5
1 − t
2
5
√
x
2
− 6x + 15
=
dx
(x − 3)
2
+ 6
=
dx
t = x − 3
dt = dx
=
√
dt
t
2
+ 6
(
)
= ln
t +
t
2
+ 6
+ C = ln
x − 3 +
x
2
− 6x + 15
+ C,
(∗) t +
t
2
+ 6 > 0
3 − 2x − x
2
dx =
4 − (x + 1)
2
dx =
t = x + 1
dt = dx
=
4 − t
2
dt
=
t
2
4 − t
2
+ 2 arcsin
t
2
+ C =
x + 1
2
3 − 2x − x
2
+ 2 arcsin
x + 1
2
+ C,
x ∈ [−3, 1]
x
2
− 2x + 5 dx =
(x − 1)
2
+ 4 dx =
t = x − 1
dt = dx
=
t
2
+ 4 dt
=
t
2
t
2
+ 4 +
4
2
ln
t +
t
2
+ 4
+ C
=
x − 1
2
x
2
− 2x + 5 + 2 ln
x − 1 +
x
2
− 2x + 5
+ C
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