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wyklady.dvi
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
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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
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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
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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
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√
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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