1. 0dx = C,
2. dx = x + C,
xn+1
3. xndx = + C, n = -1,

n+1
f (x)
1
4. dx = ln |x| + C, dx = ln |f(x)| + C
x f(x)
1
5. exdx = ex + C, eaxdx = eax + C,
a
ax
6. axdx = + C,
FUNKCJA POCHODNA
ln a
c
0
1
7. sin xdx = - cos x + C, sin(ax)dx = -a cos(ax) + C,
x
1
1
xr
8. cos xdx = sin x + C, cos(ax)dx = sin(ax) + C,
rxr-1
a
cos x
sin x
1
9. = -ctgx + C,
cos x
- sin x sin2 x
1
tgx 1
10. = tgx + +C,
cos2 x
cos2 x
1
ctg x
-
1 1 x
sin2 x
11. = arctg + C,
a2+x2 a a
ax ax � ln a
1
ex ex
"
12. = arcsinx + C,
a
a2-x2
1
ln x
x
1 13. shxdx = chx + C,
loga x
x ln a
1
"
arcsin x
14. chxdx = shx + C,
1-x2
1
arccos x - "
1
1-x2
15. = -cthx + C,
sh2x
1
arctgx
1+x2
1
16. = thx + C,
1
ch2x
arcctgx -
1+x2
"
1
sinh x cosh x "
17. = ln |x + x2 + k| + C,
x2+k
cosh x sinh x
" " "
x k
18. x2 + kdx = x2 + k + ln |x + x2 + k| + C,
2 2
" [cf(x)] = cf (x),
dx
"
19. = ln |2x + p + 2 x2 + px + q| + C,
x2+px+q
" [f(x) ą g(x)] = f (x) ą g (x)],
" "
x a2 x
" [f(x) � g(x)] = f (x)g(x) + g (x)f(x), 20. a2 - x2dx = a2 - x2 - arcsin + C,
2 2 a
1 n-1
f(x) f (x)g(x)-g (x)f(x)
21. sinn xdx = - cos x sinn-1 x + sinn-2 xdx, n e" 2,
" = . n n
g(x) (g(x))2
1 n-1
22. cosn xdx = sin x cosn-1 x + cosn-2 xdx, n e" 2.
n n
Ą Ą Ą Ą
ą 0
6 4 3 2
" "
sin x
1 2 3
1. sin2 x + cos2 x = 1, tgx = ,
sin ą 0 1 cos x
2 2 2
" "
3 2 1
cos ą
1 0 2. sin 2x = 2 sin x cos x,
2 2 2
"
"
3
tg ą -
0 1 3
3
3. cos 2x = cos2 x - sin2 x.
"
"
3
ctg ą -
3 1 0
3
b
długość łuku krzywej |L| = 1 + [f (x)]2dx,
a
b b
objętość bryły |V | = Ą f2(x)dx, |V | = 2Ą xf(x)dx.
a a
b b
pole powierzchni bryły |Ł| = 2Ą f(x) 1 + [f (x)]2dx, |Ł| = 2Ą x 1 + [f (x)]2dx.
a a
1