1. 0dx = C,
2. dx = x + C,
xn+1
3. xndx = + C,
n+1
1
4. dx = ln |x| + C,
x
5. exdx = ex + C,
ax
6. axdx = + C,
FUNKCJA POCHODNA
ln a
c
0
7. sin xdx = - cos x + C,
x
1
xr
rxr-1 8. cos xdx = sin x + C,
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
sin2 x
11. = arctgx + C,
1+x2
ax ax � ln a
1
ex ex
"
12. = arcsinx + C,
1-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 ą 1| + C,
x2ą1
cosh x sinh x
" " "
x a2
18. x2 + a2dx = x2 + a2+ ln |x+ x2 + a2|+C,
2 2
" " "
x a2
19. x2 - a2dx = x2 - a2- ln |x+ x2 - a2|+C,
2 2
"
dx
"
20. = ln |x + x2 + a2| + C,
x2+a2
"
dx
"
21. = ln |x + x2 - a2| + C,
x2-a2
" "
x a2 x
22. a2 - x2dx = a2 - x2 - arcsin + C.
2 2 a
Ą Ą Ą Ą
ą 0
6 4 3 2
" "
1. sin2 x + cos2 x = 1,
1 2 3
sin ą 0 1
2 2 2
" "
2. sin 2x = 2 sin x cos x,
3 2 1
cos ą
1 0
2 2 2
"
"
3
3. cos 2x = cos2 x - sin2 x = 1 -
tg ą -
0 1 3
3
"
"
2 sin2 x = 2 cos2 x - 1.
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