TablicaCalek,
[ Pobierz całość w formacie PDF ]Tablica calek nieoznaczonych
© 2009 by Matej
Uwaga: u, v, i w sa funkcjami zmiennej x. a, c, n, sa stalymi.
n
1
n
1
–
n
2
–
n
2
–
n
1
–
n
2
–
2
3
b
1
2
2
ax b
+
Wszystkie funkcje trygonometryczne wykorzystuja radiany.
4.
5.
1.
2.
3.
∫
csc
x
d
x
=
–
------------
csc
x
ctg
x
+
------------
∫
csc
x
d
x
1.
∫
abx
+
d
x
=
------
abx
+
(
)
32
/
2c.
∫
-----------------------------
x
d
=
–
------------------
b
2
4
ac
–
=
0
Stala musi byc dodana do wyniku kazdego calkownania.
ax
2
bx c
++
–
csc
x
d
∫
x
1
–
1
2
15
b
2
Calki zawierajace:
1.
Glowne i podstawowe calki
=
csc
+
ln
x
+
x
2
–
1
2.
∫
xa bx
+
d
x
=
----------- 3
bx
2
a
–
(
)
(
abx
+
)
32
/
a
2
±
b
2
x
2
1.
2.
3.
5.
af
()
x
d
∫
af
()
x
=
∫
d
∫
ab
1
1
–
bx
a
Funkcje z mieszanymi funkcjami trygonometrycznymi:
∫
2
x
n
(
abx
+
)
32
/
2
an
b
2
n
3
+
---------------------
x
------
=
tan
------
(
a
0
b
0
>
>
,
)
----------------------------------
------------------------
x
n
1
–
∫
uv
( )
d
∫
ux
=
d
∫
vx
±
∫
d
3.
x
n
ab
+
=
–
abx
+
d
a
2
+
b
2
x
2
∫
sin
x
cos
xd
=
(
sin
2
)
⁄
2
b
2
n
3
+
(
)
(
)
uv
d
∫
uv v u
=
g
′ ()
gx
()
–
∫
d
∫
ab
1
–
1
bx
a
1
2
ab
abx
+
abx
–
ab
–( )
cos
x
2
ab
–
ab
( )
cos
x
2
ab
+
d
∫
3
b
2
-------------------
x
x
abx
+
--------
bx
2
a
–
2.
---------------------
=
------
tanh
------
=
---------
ln
--------------
∫
sin
ax
cos
bx d
=
–
-----------------------------
–
-----------------------------
4.
=
(
)
abx
+
(
)
(
)
a
2
–
b
2
x
2
------------
x
d
∫
g
()
=
ln
dla
(
a
0
b
0
>
>
,
)
xx
sec
d
∫
x
tg
=
sec
4.
∫
csc
x
ctg
x x
d
=
–
csc
x
∫
2
x
n
abx
x
n
abx
+
+
b
2
n
1
+
------------------------
x
n
1
–
abx
+
-------------------
x
----------------------------
∫
-------------------
x
d
5.
=
–
{
[
gx
()
]
⁄
g
()
r
}
(
r
1
+
)
r
1
≠
(
)
(
)
∫
a
2
2
b
bx
a
1
b
2
x
2
a
2
r
1
+
m
1
+
n
1
–
sin
cos
mn
+
x
x
n
1
–
mn
+
3.
a
2
b
2
x
2
+
x
=
---
a
2
b
2
x
2
+
+
------
ln
------
+
+
----------
6.
∫
[
g
()
]
r
g
′
()
x
d
=
5a.
∫
sin
m
cos
n
d
x
=
--------------------------------------
+
-------------
∫
sin
m
cos
n
2
–
x
d
ln
=
1
1
–
abx
+
a
-------
l
n
---------------------------------
dla
(
a
0
>
)
1
1
---
bx
a
1
b
2
x
2
a
2
m
1
–
n
1
+
x
n
1
+
n
1
+
sin
cos
mn
+
x
x
m
1
–
mn
+
4.
∫
--------------------------
x
d
=
ln
------
+
+
----------
m
n
--------------------------------------
-------------
m
2
–
n
a
+
a
7.
∫
x
n
d
x
=
------------
(
n
≠
–
1
)
∫
sin
cos
x
d
=
–
+
∫
sin
x
cos
x
d
x
1
xa bx
+
6.
∫
----------------------
d
=
a
2
+
b
2
x
2
2
1
–
abx
+
∫
a
2
2
b
–
1
bx
a
d
x
----------
tan
---------------
dla
(
a
0
<
)
1
a x
sin
1
ca
2
1
---
cx
1
–
+
b
---
---
a
2
------
8.
-----
∫
x
1
=
–
d
∫
x
=
ln
(
x
0
≠
)
------------------------------------------
x
d
-----------------------
5.
a
2
–
b
2
x
2
x
=
–
b
2
x
2
+
sin
------
6.
∫
=
ln
tg
tg
–
a
–
a
x
+
b x
cos
+
b
2
1
1
---
1
–
bx
a
-----
∫
x
n
–
d
∫
x
1
n
d
x
n
x
–
1
n
–
6.
Calki zawierajce:
1.
∫
-------------------------
x
d
=
sin
------
9.
=
=
------------
(
n
1
≠
)
1
abx
+
an
1
–
b
2
n
3
–
(
)
1
7.
∫
------------------------
x
d
=
–
------------------------------
–
-----------------------
∫
------------------------------
x
d
Calki zawierajace
e
x
e
ax
a
2
–
b
2
x
2
x
n
a
b
x
+
(
)
x
n
1
–
2
an
1
–
(
)
x
n
1
–
abx
+
SIN
1.
1
---
Fu
()
u
∫
Fe
ax
()
d
=
-----------
u
∫
u
,
=
e
ax
abx
+
x
1
xa bx
+
2
ax x
2
–
sin
d
∫
---
8.
-------------------
x
∫
2
abx
=
+
+
a
∫
----------------------
x
d
x
sin
x
d
∫
x
=
–
cos
2.
ax
=
–
cos
ax
1
---
e
ax
–
1
∫
xa
–
2
a
2
-----
x
---
1.
e
x
d
∫
e
x
=
2.
∫
e
ax
x
d
=
3.
xe
x
x
d
∫
xe
x
e
x
=
–
2
ax x
–
=
-----------
2
ax x
2
–
+
cos
1
d
∫
abx
abx
+
x
n
(
+
)
32
/
b
2
n
5
–
(
)
abx
+
x
n
1
–
sin
∫
---
x
1
---
2
x
d
∫
---
x
1
---
2
ax
sin
-------------------
x
------------------------------
-----------------------
∫
-------------------
x
d
2
2
-----------------
9.
=
–
–
3.
d
x
=
–
sin
4.
sin
ax
x
=
–
xe
ax
d
∫
e
ax
a
2
an
1
–
(
)
x
n
1
–
2
an
1
–
(
)
a
–
1
4.
=
------
ax
1
–
(
)
5.
x
n
e
x
x
d
∫
x
n
e
x
nx
n
–
e
x
x
=
–
∫
d
∫
2
x
2
–
ax
–
3
a
2
a
3
-----
x
---
----------------------------------
2
ax x
2
–
2.
x
2
ax x
2
–
x
=
+
cos
1
–
1
1
---
3
---
xx
3
---
x
1
ax b
+
2
ac
cax b
+
( )
acx d
+
6
4
3
-------------------
---
-----------------
x
d
----------
5.
∫
sin
d
x
=
–
sin
cos
x
–
sin
cos
+
10.
∫
=
tanh
-----------------------
d
ab
cx
+
x
x
---
1
ac
(
)
------
ab
cx
+
cx d
+
6.
∫
-------------------
=
–
ln
(
)
–
1
2
ax x
2
–
x
---
3.
d
∫
2
ax x
2
------------------------
x
=
–
+
a
cos
1
1
---
n
1
–
n
n
n
1
–
n
2
–
x
6.
7.
8.
9.
10.
11.
12.
=
xx
sin
x
d
∫
x
sin
d
x
–
sin
x
cos
x
+
------------
∫
sin
x
d
a
x
+
b
ax
b
ab
–
b
Calki zawie
r
ajace:
a
2
–
x
2
a
2
–
x
2
7.
∫
=
ab
x
ce
2
x
++
d
------
–
------------
b
x
+
ln
(
)
–
1
d
∫
22
ax x
2
2
ax x
2
–
–
x
---
+
x
=
x x
sin
sin
–
xx
cos
4.
------------------------
x
=
–
----------------------------
–
cos
1
x
∫
aFa u
F
(
2
–
x
2
)
=
u
cos
∫
x
(
cos
)
,
=
au
sin
x
2
x
∫
a
d
------
ce
x
abd
–
+
cd
2
8.
---------------------------------
x
=
+
–
------------------------------
de
x
+
ln
(
)
d
∫
ax
x
=
--------------
x x
–
------------------
cos
a
+
x
d
1
a
2
x
2
–
------
xa
+
xa
–
1
–
1.
∫
---------------
x
d
=
ln
1
2
ax x
2
–
x
---
------------
------------------------
x
d
5.
∫
=
cos
1
x
e
x
1
–
1
---
e
x
1
+
e
x
1
–
9.
∫
--------------------
x
d
=
–
-------------
x
2
x
sin
d
∫
x
2
=
–
cos
x
+
2
xx
sin
+
2
cos
x
(
)
2
x
---
a
2
a
2
-----
x
--
2.
∫
a
2
–
x
2
d
x
=
–
x
2
+
arcsin
x
2
ax x
2
–
–
1
x
---
6.
------------------------
x
d
∫
2
ax x
2
=
–
–
+
a
cos
1
d
∫
x
n
x
n
nx
n
1
–
x
sin
x
=
–
cos
x
+
∫
c
o
s
xx
d
∫
--- 1
e
ax
1
---
1
e
ax
+
+
1
10.
1
e
ax
+
x
=
+
+
ln
-----------------------------
1
---
a
2
1
e
ax
+
–
1
3.
∫
xa
2
–
x
2
d
x
=
–
(
–
x
2
)
32
/
d
∫
x
x
–
1
arcsin
x x
=
arcsin
+
1
–
x
2
d
∫
x
3
a
x
2
2
ax x
2
–
+
2
3
a
2
2
x
---
1
2
---
x
---
7.
------------------------
x
=
–
---------------
2
ax x
2
–
+
--------
cos
1
11.
∫
--------------------
x
d
=
ln
(
e
x
+
a
2
–
a
)
–
1
---
1
a
2
x
2
x
--- 2
x
2
a
4
-----
x
--
∫
x
1
–
1
–
sin
ax
x
=
(
sin
a x
)
+
–
e
x
+
a
2
4.
∫
x
2
a
2
–
x
2
d
x
=
(
–
a
2
)
a
2
–
x
2
+
arcsin
1
x
2
ax x
2
–
2
ax x
2
–
ax
∫
---------------------------
x
d
------------------------
8.
=
–
----------------
∫
---
d
x
13.
=
–
ctg
ax
Calki zawierajace
a
x
x
n
a
2
–
x
x
2
a
a
2
x
2
+
–
5.
--------------------
x
d
∫
a
2
=
–
x
2
–
ln
sin
ax
2
-----------------------------
d
∫
a
x
d
∫
x
n
a
x
a
n
1
2
ax x
2
–
xa
–
x
1.
a
x
x
=
--------
2.
x
n
a
x
x
=
----------
–
--------
x
n
1
–
∫
a
x
d
x
9.
Calki rozne
:
1.
∫
------------------------------
x
d
=
------------------------------
sin
2
ab
+
(
ab
+
)
x
sin
2
ab
–
(
ab
–
)
x
ln
a
ln
a
ln
(
)
32
/
14.
∫
sin
ax
sin
bx
d
x
=
–
----------------------------
+
----------------------------
a
2
2
ax x
2
–
a
2
x
2
–
x
2
1
---
a
2
x
--
(
)
(
)
x
n
x
3
1
+
2
2
n
1
–
---------------
x
n
3
–
x
3
1
+
6.
∫
--------------------
x
d
=
–
–
x
2
–
arcsin
3.
∫
------------------
x
d
=
---------------
x
n
2
–
x
3
+
1
–
∫
------------------
d
∫
--- ---
a
2
1
–
15.
COS
1.
-----------------------
x
=
tg
------
1
a
2
x
--
d
∫
2
ax
x
ax
+
a
x
–
+
2
a
2
2
1
–
x
---
1
+
sin
ax
x
n
x
4
x
n
3
–
n
1
–
n
3
–
n
1
–
7.
--------------------
x
∫
=
x
a
2
arcsin
------------
x
=
–
---------------
a
2
x
2
–
+
-----
sin
------------
x
n
4
–
4.
∫
------------------
x
d
=
------------
x
4
+
1
–
-----------------
∫
d
x
–
x
2
+
1
x
4
+
1
∫
---
x
32
x
x
3
a
–
1
---
ax
8.
--------------------
x
d
∫
a
2
x
2
=
–
–
2.
-------------
=
ln
(
/
+
x
3
–
a
)
x
cos
x
d
∫
x
=
sin
2.
∫
cos
ax x
d
=
sin
Calki zawierajace
ln
x
()
–
x
2
3.
∫
2
cos
x
d
=
1
---
x
+
1
--- 2
x
sin
4.
∫
cos
2
ax
d
x
=
1
---
x
+
1
---
sin
a
2
ax
d
∫
F
()
e
u
Fx
ln( )
x
=
∫
u
,
=
ln
x
9.
d
∫
--------------------
x
x
2
a
2
=
–
---
a
2
–
x
2
+
a
2
-----
arcsin
x
--
3.
-
-----------
x
d
∫
x
2
=
1
x
2
+
1
x
2
–
x
xa
+
+
ax
–
a
ln
(
x a
+
+
x
)
1.
ln
∫
x
xd
=
–
+
xx
ln
2.
ln
∫
x
ax d
=
–
+
xax
ln
–
x
2
1
---
3
---
3
---
x
1
2
2
x
+
1
x
2
–
1
x
4
+
5.
∫
4
cos
x
d
=
cos
3
sin
x
+
cos
xx
sin
+
d
∫
---
a
1
xa
2
+
2
–
x
2
4.
∫
-------------------------------------
d
=
-------
ln
-----------------------------------
d
∫
xx
∫
10.
-----------------------
x
=
–
ln
3.
4.
x
l( )
n
x
=
ln( )
n
–
nx
ln( )
n
1
–
d
x
-----------------------------
–
x
2
x
(
)
1
x
4
+
1
---
x
1
xx
ln
6.
7.
8.
9.
10.
11.
12.
=
xxx
d
cos
∫
x
n
cos
x
cos
x
n
1
–
sin
x
+
n
1
–
n
∫
cos
x
n
2
–
d
x
d
∫
x
=
ln
5.
d
∫
x
-----------
x
=
ln
ln
11.
d
∫
a
2
x
1
x
2
a
2
x
2
–
=
–
--------
a
2
–
x
2
d
∫
n
1
+
1
n
1
+
=
x xx
d
cos
+
xx
sin
6.
x
n
x
ln
x
=
------------
x
n
1
+
ln
x
–
-------------------
x
n
1
+
x
--- 2
x
2
3
a
4
8
x
--
(
)
2
arcsin
---------------
x x
cos
a
2
ax
sin
a
12.
∫
(
a
2
–
x
2
)
32
/
d
=
–
(
–
5
a
2
)
a
2
–
x
2
+
--------
∫
cos
=
+
-----------------
e
ax
7.
∫
e
ax
sin
bx
d
x
=
----------------
a x
sin
(
–
b x
cos
)
a
2
+
b
2
1
x
POCHODNA FUNKCJI
13.
∫
--------------------------
x
d
=
-------------------------
cos
∫
x
2
x
2
xx
d
=
sin
x
+
2
xx
cos
–
2
x
sin
e
ax
a
2
(
a
2
–
x
2
)
32
/
a
2
a
2
–
x
2
1
–
8.
∫
e
ax
cos
bx x
d
=
----------------
ab
cos
xb bx
sin
(
–
)
dw
dx
-------
du
dx
du
dx
dx
du
1
dx du
⁄
cos
∫
x
n
x
n
nx
n
1
–
+
b
2
-------
=
------
: Regula lancuchowa
------
=
-----
=
----------------
xd
=
sin
x
–
∫
x
sin
d
Calki zawierajce
x
2
±
a
2
x
2
±
a
2
–
cos
x
d
∫
x
1
–
1
Funkcje hiperboliczne
=
cos
–
1
x
2
–
d
dx
du
dx
d
du
d
dx
du
dx
dv
dx
–
1
x
--
------
f
()
=
------
------
f
()
------
uv
+
(
)
=
------
+
------
e
x x
–
–
2
e
x
+
2
e
–
x
sinh
x
Uwaga:
ln
x
+
x
2
+
a
2
=
sinh
Uwaga:
sinh
x
=
-----------------
cosh
x
=
-----------------
tgh
x
=
--------------
1
--- 1
a
2
x
2
–
cos
x
∫
x
1
1
–
ax
=
(
cos
a x
)
–
–
cosh
x
------
u
()
u
dv
dx
------
v
du
dx
------
u
--
1
v
2
----
v
du
dx
------
u
dv
dx
–
------
1
---
1
---
x
=
+
------
=
–
1
x
--
1
–
a
--
2
sinh
d
ax
2
a
2
+
+
1.
x
sinh
x
d
∫
x
=
cosh
2.
∫
x
=
sinh
2
x
–
1
---
ln
x
+
x
2
a
2
–
=
cosh
,
ln
=
sinh
d
x
--
---------------------------
13.
∫
-----------------
=
tg
ax
x
2
d
dx
()
vu
v
–
du
dx
u
dv
dx
cos
ax
1
---
1
---
x
2
cosh
x
d
------
u
v
=
------
u
v
+
ln
------
3.
x
cosh
x
d
∫
x
=
sinh
4.
∫
x
=
sinh
2
x
+
( )
2
sec
u
∫
x
a
2
(
)
x
∫
aFa u
F
2
=
sec
,
=
au
tan
sin
2
ab
+
(
ab
+
)
x
sin
2
ab
–
(
ab
–
)
x
14.
∫
cos
ax
cos
bx x
d
=
----------------------------
+
----------------------------
(
)
(
)
d
dx
1
---
d
dx
5.
6.
7.
∫
tgh
x x
d
=
ln
(
cosh
x
)
a
2
( )
x
∫
aFa u
F
2
=
∫
(
tg
)
sec
u
u
tan
u
d
,
x
=
au
sec
------
x
ln
=
------
e
x
=
e
x
∫
---
1
ax
2
------------------------
x
15.
=
tan
------
–
1
1
+
cos
ax
x x
d
∫
x
ctgh
=
ln
sinh
d
∫
---
----------------
x
1
x
--
d
∫
2
a
---------------
x
1
------
xa
–
xa
+
d
dx
d
dx
()
nu
n
–
du
dx
1.
=
tan
2.
=
ln
------------
x
2
+
a
2
x
2
–
a
2
------
a
x
=
a
x
ln
a
------
u
n
=
------
2
sech
x
d
∫
x
∫
x
sech
d
=
arctg
(
sinh
x
)
8.
=
tgh
sin
x
d
∫
---
x
2
x
d
(
log
a
)
TG
tg
=
-----------
----------------
x
1
---
du
dx
3.
=
ln
+
a
2
---------------------
=
(
log
a
)
------
cos
x
9.
10.
11.
∫
x
sech
x x
d
=
–
sech
x
x
2
+
a
2
dx
1.
xx
d
∫
x
tg
=
–
ln
cos
2.
∫
tg
ax x
d
=
–
1
---
ln
cos
ax
)
∫
csch
x
ctgh
x x
d
=
–
csc
h
x
4.
∫
x
2
±
a
2
x
=
---
x
2
±
a
2
±
-----
x
ln
+
x
2
±
a
2
dx
sin
(
)
⁄
dx
=
cos
x
d
(
cos
x
)
⁄
dx
=
–
sin
x
1
---
x
2
2
1
---
∫
x
csch
x
d
=
ln
tgh
1
---
x
2
dx
tan
(
)
⁄
dx
=
sec
dx
cot
(
)
⁄
dx
=
–
csc
2
2
3.
x
d
∫
x x
tg
x
=
tg
–
4.
x
d
∫
x
tg
ax
=
–
+
tg
ax
5.
∫
xx
2
±
a
2
d
x
=
(
±
a
2
)
32
/
dx
sec
(
)
⁄
dx
=
sec
x
tan
x
dx
csc
(
)
⁄
dx
=
–
csc
x
cot
x
e
ax
a
2
b
2
–
–
1
n
1
n
1
–
n
1
–
n
2
–
12.
∫
e
ax
sinh
bx
d
x
=
----------------
a x
sinh
(
–
b
cosh
b x
)
x
--- 2
x
2
a
4
-----
x
d
(
sin
)
------------------
d
u
dx
1
1
u
2
–
---
1
–
sin ---
5.
∫
tg
x
d
x
=
------------
tg
x
–
∫
tg
x
d
x
-----------------------
=
------
–
≤
u
≤
6.
∫
x
2
x
2
±
a
2
d
x
=
(
±
a
2
)
x
2
±
a
2
–
ln
+
x
2
±
a
2
dx
dla
e
ax
a
2
b
2
–
–
d
∫
x
1
–
1
1
---
x
2
13.
∫
e
ax
cosh
bx
d
x
=
----------------
a
(
cosh
b x
–
b x
sinh
)
x
2
+
x
a
2
–
1
a
--
–
1
6.
tg
=
tg
x
–
ln
(
+
1
)
d
(
cos
)
------------------
d
u
dx
–
1
u
2
–
1
7.
--------------------
x
d
∫
x
2
a
2
=
+
–
a
sinh
–
1
------------------------
=
------
(
0
≤
cos
≤
π
)
dx
dla
1
1
---
x
1
---
7.
∫
-----------------------
x
d
=
+
ln
(
cos
ax
+
sin
ax
)
Calki zawierajace
abx
+
x
2
–
x
a
2
–
1
1
–
x
-
--
d
(
tan
)
--------------
du
dx
---
---
1
+
tg
ax
d
∫
x
2
--------------------
x
1
---
Fu
8.
=
–
a
2
–
a
sec
–
1
-----------------------
=
------
dla
–
<
tan
u
<
∫
Fa bx
+
(
)
d
x
=
()
u
∫
u
,
=
abx
+
dx
cos
x
1
x
2
x
2
a
2
x
2
x
a
2
–
1
CTG
ctg
x
=
-----------
=
-----------
d
∫
---
abx
1
abx
+
9.
∫
--------------------
x
d
=
–
--------------------
+
ln
x
+
x
2
±
a
2
d
(
cot
)
--------------
du
dx
1
1
–
1.
---------------
x
=
ln
+
-----------------------
=
------
dla
(
0
<
cot
u
<
π
)
sin
x
tg
x
dx
1.
x x
d
∫
x
ctg
=
ln
sin
2.
d
∫
---
ctg
ax x
=
ln
(
sin
ax
)
1
abx
+
–
ba bx
+
1
1
x
2
–
1
2.
∫
----------------------
x
d
=
-----------------------
10.
--------------------
x
d
∫
x
=
ln
+
x
2
±
a
2
-----------------------
(
sec
)
---------------------
d
u
dx
------
≤
–
sec
---
1
–
1
---
=
0
<
–
π
≤
sec
<
–
(
)
2
(
)
±
a
2
dx
1
---
d
∫
x
2
d
∫
x
2
1
abx
+
–
1
3.
ctg
x
x
=
–
ctg
–
x
4.
ctg
ax
x
=
–
–
ctg
ax
3.
∫
----------------------
x
d
=
------------------------------------------------
(
n
1
≠
)
d
∫
x
2
-------------------------
x
1
−
--------------------
±
a
2
x
a
2
–
1
11.
=
d
(
csc
)
---------------------
d
u
dx
1
–
csc
---
---
(
)
n
(
n
1
–
)
ba bx
+
(
)
n
1
–
-----------------------
------
1
–
1
=
≤
<
0
–
π
<
csc
≤
–
x
2
x
2
±
a
2
dx
d
∫
n
1
–
–
1
n
------------
n
1
–
∫
n
2
–
x
abx
+
1
b
2
5.
ctg
x
x
=
–
ctg
x
–
ctg
x
d
4.
∫
---------------
x
d
=
-----
abxa
a
bx
+
[
+
–
ln
]
1
xx
2
a
2
+
1
---
–
1
a
--
12.
∫
-----------------------
d
=
–
sinh
d
(
sinh
x
)
⁄
dx
=
cosh
x
d
(
cosh
x
)
⁄
dx
=
sinh
x
1
---
x
2
–
d
∫
x
1
–
1
2
2
6.
ctg
=
ctg
x
+
ln
(
+
1
)
x
abx
+
1
b
2
a
abx
+
d
(
tanh
x
)
⁄
dx
=
sech
x
d
(
coth
x
)
⁄
dx
=
–
csch
5.
∫
----------------------
x
d
=
-----
---------------
+
ln
a
bx
+
1
xx
2
1
---
–
1
x
-
--
(
)
2
13.
∫
-----------------------
x
d
=
sec
d
(
sech
x
)
⁄
dx
=
–
sech
x
tanh
x
–
a
2
1
x
abx
+
a
2
bx
+
2
b
2
abx
+
d
(
csch
x
)
⁄
dx
=
–
csch
x
coth
x
SEC
1.
2.
3.
4.
sec
x
=
-----------
6.
∫
----------------------
x
d
=
–
-------------------------------
x
x
2
c
o
s
x
(
)
(
)
d
∫
x
2
--------------------
x
d
dx
------------------
d
u
dx
1
u
2
3
2
14.
=
±
a
2
–
1
±
a
2
------
sinh
=
------
x
sec
x
d
∫
x
=
ln
sec
+
tg
x
x
2
abx
+
1
b
3
1
---
abx
+
1
7.
∫
---------------
x
d
=
-----
(
+
)
2
–
2
aa bx
+
(
)
+
a
2
ln
a
bx
+
x
2
x
2
a
2
±
x
---
x
2
a
2
-----
x
−
15.
16.
∫
--------------------
x
d
=
±
a
2
ln
+
x
2
±
a
2
d
dx
------------------
d
u
dx
1
u
2
1
---
------
–
1
------
–
1
cosh
=
dla
u
1
>
i
cosh
u
≥
0
∫
sec
ax
d
x
=
ln
(
sec
ax
+
tg
ax
)
–
1
x
2
abx
+
-----
abx
a
2
abx
+
8.
∫
----------------------
x
d
=
+
–
---------------
–
2
a
a
x
+
ln
d
dx
--------------
du
dx
2
sec
d
∫
x
(
)
2
x
--- 2
x
2
3
a
4
8
x
=
tg
–
1
∫
(
x
2
±
a
2
)
32
/
d
x
=
(
±
5
a
2
)
x
2
±
a
2
+
--------
x
ln
+
x
2
±
a
2
------
tanh
u
=
------
dla
–
1
u
1
<<
(
abx
+
)
n
1
+
3
sec
d
∫
---
1
---
9.
∫
(
abx
+
)
n
d
=
----------------------------
(
n
≠
–
1
)
x
=
sec
xx
tan
+
ln
s
ec
x
+
tan
x
bn
1
+
(
)
17.
∫
--------------------------
x
d
1
=
±
-------------------------
x
1
xa bx
+
1
---
(
x
2
±
a
2
)
32
/
x
a
bx
+
a
2
x
2
±
a
2
1
n
1
–
n
2
–
n
1
–
10.
∫
-----------------------
x
d
=
ln
5.
6.
∫
n
sec
d
x
=
------------
sec
n
2
–
x
tan
x
+
------------
∫
sec
n
2
–
x
d
x
(
)
---------------
1
1
ax
-----
abx
+
x
Calki zawierajace
1.
ax
2
bx c
++
–
sec
x
d
∫
x
1
–
1
=
sec
–
ln
x
+
x
2
–
1
11.
∫
-------------------------
x
d
=
–
------
+
ln
---------------
1
ax
2
c
+
1
ac
1
–
x
2
(
abx
+
)
∫
-----------------
x
d
=
----------
tan
--
x
a
1
1
xa bx
+
-----------------------
1
a
2
CSC
1.
2.
3.
csc
x
=
----------
12.
Calki zawie
r
ajace:
∫
-------------------------
d
=
+
-----
ln
x
a
bx
+
sin
x
(
)
2
(
)
1
2
4
ac b
2
–
1
–
2
ax b
+
4
ac b
2
–
2a.
∫
-----------------------------
d
=
------------------------
tan
-----------------------
ax
2
bx c
++
xd
csc
∫
x
=
–
ln
csc
+
ctg
x
abx
+
1
---
dla
4
ac b
2
–
>
0
∫
csc
ax d
=
ln
(
csc
ax
–
ctg
ax
)
∫
---
uF
()
u
Fabx
+
(
)
x
=
∫
u
,
=
abx
+
1
1
b
2
4
ac
–
2b.
∫
-----------------------------
x
d
=
------------------------
ln
--------------------------------------------------
–
b
2
–
4
ac
ax
2
bx c
++
csc
∫
x
2
++
b
2
4
ac
–
x
d
=
–
ctg
∫
---
u
n
1
Fabx
+
(
n
)
x
=
∫
u
–
F
()
u
,
=
n
abx
+
dla
b
2
4
ac
–
>
0
UWAGA
cos
–
1
oznacza to samo co arccos itp
2
an
b
2
n
1
+
abx
+
∫
-------------
x
d
sin
a
2
1
2
a
2
n
4
–
2
n
1
–
-----------------
d
∫
---
------------
-------------------------
x
dw
du
d
dx
d
dx
(
a
2
2
Calki zawiera
1
1
u
2
+
–
1
u
2
+
1
uu
2
1
–
d
–
uu
2
1
b
3
1
1
u
2
–
b
a
2
1
aa bx
+
---------------
2
ax b
+
2
ax b
[ Pobierz całość w formacie PDF ]