Podstawowe wzory całek
1. x C
dx k
xk k +
= + +1
1
1 , k≠−1
2. dx x C
x = ln + 1
3. a C
dx a
ax = x+ ln
1
4. e xdx e x+C
=β β
β 1
5. xdx αx +C
=α
α 1sin( )
) cos(
6. xdx αx +C
−α
=
α 1cos( )
) sin(
7. =
x dx
cos x C
x
− + + 1 tan
1 lntan
2
2 C
x
x +
−
= +
1 sin
1 lnsin
21
8. dx x C
x = tg + cos
1
2
9. =
x dx
sin lntan2x +C
10. dx x C
x = ctg− + sin
1
2
11. C
a x x
a
dx = +
− arcsin| |
2 2
12. C
a x a a
x
dx = +
+ 1arctg
2 2
13. C
a k x a a
k x
dx = + +
+
+ 1arctg
)
( 2 2
14. lnxdx=(x−1)⋅lnx+C 15. lnnxdx=xlnnx−n lnn−1xdx+C
16. =− − ⋅ + − − xdx
n x n n x
dx
x n n
n 1 1 sin 2
cos 1sin
sin
17. = − ⋅ + − − xdx
n x n n x
dx
x n n
n 1 1 cos 2
sin 1cos
cos
18. arcsinxdx=x⋅arcsinx+ 1−x2+C 19. arccosxdx=x⋅arccosx− 1−x2+C 20. arctanxdx=x⋅arctanx−21ln1+x2 +C 21. arccotx dx=x⋅arccotx+21ln(1+x2)+C
22. ; 2
) 1 2 (
2 3 2 ) 1 2 ( 2
1 ) 1
( 2 2 1 2 1+ ≥
− + + −
⋅ +
= −
+ − − C n
x dx n
n x
x x n
dx
n n
n
23. x x k C
k x
dx = + + +
+
2
2 ln
24. x2+k dx=12 x x2+k+klnx+ x2+k +C
25. C
a a x
x a x dx x
a − = − + +
| arcsin|
2 2 2 21 2 2
